Can every proof by contradiction also be shown without contradiction? Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantages/disadvantages of proving by contradiction?

As an aside, how is proving by contradiction viewed in general by 'advanced' mathematicians. Is it a bit of an 'easy way out' when it comes to trying to show something or is it perfectly fine? I ask because one of our tutors said something to that effect and said that he isn't fond of proof by contradiction.
To determine what can and cannot be proved by contradiction, we have to formalize a notion of proof.  As a piece of notation, we let $\bot$ represent an identically false proposition. Then $\lnot A$, the negation of $A$, is equivalent to $A \to \bot$, and we take the latter to be the definition of the former in terms of $\bot$. 

There are two key logical principles that express different parts of what we call "proof by contradiction":


The principle of explosion: for any statement $A$, we can take "$\bot$ implies $A$" as an axiom.  This is also called ex falso quodlibet.  
The law of the excluded middle: for any statement $A$, we can take "$A$ or $\lnot A$" as an axiom. 


In proof theory, there are three well known systems:


Minimal logic has neither of the two principles above, but it has basic proof rules for manipulating logical connectives (other than negation) and quantifiers. This system corresponds most closely to "direct proof", because it does not let us leverage a negation for any purpose. 
Intuitionistic logic includes minimal logic and the principle of explosion
Classical logic includes intuitionistic logic and the law of the excluded middle


It is known that there are statements that are provable in intuitionistic logic but not in minimal logic, and there are statements that are provable in classical logic that are not provable in intuitionistic logic. In this sense, the principle of explosion allows us to prove things that would not be provable without it, and the law of the excluded middle allows us to prove things we could not prove even with the principle of explosion.  So there are statements that are provable by contradiction that are not provable directly. 

The scheme "If $A$ implies a contradiction, then $\lnot A$ must hold" is true even in intuitionistic logic, because $\lnot A$ is just an abbreviation for $A \to \bot$, and so that scheme just says "if $A \to \bot$ then $A \to \bot$". But in intuitionistic logic, if we prove $\lnot A \to \bot$, this only shows that $\lnot \lnot A$ holds. The extra strength in classical logic is that the law of the excluded middle shows that $\lnot \lnot A$ implies $A$, which means that in classical logic if we can prove $\lnot A$ implies a contradiction then we know that $A$ holds. In other words: even in intuitionistic logic, if a statement implies a contradiction then the negation of the statement is true, but in classical logic we also have that if the negation of a statement implies a contradiction then the original statement is true, and the latter is not provable in intuitionistic logic, and in particular is not provable directly.
Following Carl Mummert in considering the three main systems of propositional logic, let's re-interpret the question once again as 


  Does there exist a proof by contradiction that is valid in Classical Logic, yet invalid in Minimal Logic (resp. Intuitionistic Logic)? 


The systems of Minimal Logic, Intuitionistic Logic and Classical Logic are three systems of propositional logic of strictly increasing strength. (I will be using the textbook 'Foundations of logic and mathematics' by Nievergelt as a reference, especially Sections 1.1 and 4.1.) To begin to answer this question, we first need to formalise what 'proof by contradiction' is, as a logical principle. Let us look at two examples. 

Take first the usual proof of the infinitude of primes: Suppose $p_1, \ldots, p_N$ is the list of all primes. Then the smallest prime factor of $p_1 \cdots p_N + 1$ is larger than $p_N$. Thus there are infinitely many primes. The underlying logical principle applied at the word 'thus' is the so-called Law of Reductio Ad Absurdum:
$$(P \to Q) \to ((P \to \neg Q) \to \neg P),$$
where $P$ is '$p_1, \ldots, p_N$ is the list of all primes' and Q is 'the smallest prime factor of $p_1 \cdots p_N + 1$ is a prime larger than $p_N$'. So if the Law of Reductio Ad Absurdum is valid, then the proper conclusion of the above proof is that $p_1, \ldots, p_N$ is not the list of all primes, which is reasonable as a possible definition of the infinitude of primes (where the prefix 'in-' means 'not', so that 'in-finite' means 'not-listable').    

There is another kind of proof of contradiction, namely of the Pigeonhole Principle: Given $n$ holes, if $n + 1$ pigeons are put into them, then there must be some hole with at least two pigeons. So the proof goes: Were there no hole having at least two pigeons, then at most $n$ pigeons were put into the $n$ holes. Thus, if $n + 1$ pigeons were put into the $n$ holes, then there is some hole with at least two pigeons. And the underlying logical principle at the word 'thus' is now the so-called Converse Law of Contraposition:
$$(\neg P \to \neg Q) \to (Q \to P),$$
where $P$ is 'there is some hole with at least two pigeons' and $Q$ is '$n + 1$ pigeons are put into the holes'.

By these two examples, I hope that the reader sees and is convinced that what is generically regarded as 'proof by contradiction' is formalisable as either the the Law of Reductio Ad Absurdum or the Converse Law of Contraposition, which are two separate laws distinct from each other.  

The subtlety now arises that in fact


the Law of Reductio Ad Absurdum is valid in Minimal Logic, in Intuitionistic Logic and in Classical Logic.
The Converse Law of Contraposition is not valid in Minimal Logic (resp. Intuitionistic Logic). However, adding the Converse Law of Contraposition to Minimal Logic (resp. Intuitionistic Logic) gives a logic equivalent to the full Classical Logic (see Appendix).


So finally, we can arrive at an answer to the re-interpreted question in the yellow box. For our first example, the proof of the infinitude of primes uses 'proof by contradiction' in the sense of the Law of Reductio Ad Absurdum. This proof is valid in Classical Logic, but by (1), is also valid in Minimal Logic and in Intuitionistic Logic. However, for our second example, the proof of the Pigeonhole Principle uses 'proof by contradiction' in the sense of the Converse Law of Contraposition. Although this proof is valid in Classical Logic, by (2), it is not valid in Minimal Logic nor in Intuitionistic Logic. So we must be careful not to reject as non-intuitionistic or non-minimalistic those proofs in classical mathematics that uses the Law of Reductio Ad Absurdum, and inspect carefully whether it is this law or the Converse Law of Contraposition that is being employed.  



Appendix: 

For the convenience of the reader, I write down the axioms of these three systems of logic, as taken from Nievergelt's book. One of the purposes of writing this down is that in @Carl Mummert's answer, he uses a constant symbol $\bot$ to denote the falsum. However, it is possible avoid the falsum and to write down the axioms of Minimal Logic, Intuitionistic Logic and Classical Logic completely over the language $\{\neg, \to, \vee, \wedge\}$, with the symbol $\neg$ for negation, the symbol $\to$ for implication, the symbol $\vee$ for disjunction and the symbol $\wedge$ for conjunction. In this language, the use of a constant symbol $\bot$ for the falsum is avoided. 

To give the details, let $CL^-$ be the system consists of the following two axiom schemas:


$P \to (Q \to P)$
$(P \to (Q \to R)) \to ((P \to Q) \to (P \to R))$


Then Classical Logic (CL) is $CL^-$ together with the Converse Law of Contraposition (p.58).

Let $T$ denote $CL^{-}$ together with the additional five axiom schemas: 


$(P \wedge Q) \to P$
$(P \wedge Q) \to Q$
$P \to (Q \to (P \wedge Q)$
$P \to (P \vee Q)$
$Q \to (P \vee Q)$
$(P \to R) \to ((Q \to R) \to ((P\vee Q) \to R))$


Then Minimal Logic (ML) is $T$ plus the Law Of Reductio Ad Absurdum (p.228). And, Intuitionistic Logic (IL) is $T$ plus the Special Law of Reductio Ad Absurdum -- $$(P \to \neg P) \to \neg P$$
and also plus the Law of Denial Of the Antecedent 
$$\neg P \to (P \to Q).$$

The facts are $ML + \text{Law of Denial Of the Antecedent} \Leftrightarrow IL$ (Exercise 755, p.231), $ML + \text{Law of Double Negation} \Leftrightarrow CL$ (Theorem 653, p.229) and $IL + \text{Law of Double Negation} \Leftrightarrow CL$ (Exercise 754, p.231). Since the Law of Reductio Ad Absurdum is an axiom of ML, hence it is valid in both $IL$ and $CL$. Next, $ML$ is strictly weaker than $IL$ since the Law of Denial of the Antecedent is not valid in $ML$. And also $IL$ is strictly weaker than $CL$ since the Law of Double Negation is not valid in $IL$. Hence, the Law of Double Negation being the special case of the Converse Law of Contraposition by taking $Q = \top$ as the verum, the Converse Law of Contraposition is not valid in $ML$ nor in $IL$.