Prove that $\tau=\{ \mathbb{R},\emptyset\} \cup \{(-t,t)|t\in\mathbb{I}^+\}$,is not a topology on $\mathbb{R}$. Prove that $\tau=\{ \mathbb{R},\emptyset\} \cup \{(-t,t) \,\, | \,\, t\in\mathbb{I}^+\}$ is not a topology on $\mathbb{R}$, where $\mathbb{I}^+$ denotes the positive irrational numbers.
Any pointers on how to do that?

Assuming that your topology is $\tau = \{ \mathbb R , \emptyset \} \cup \{ (-t,t) : t \in \mathbb Q^c_+ \}$
Let $t_n$ be an increasing sequence of positive irrational numbers converging to a rational number, say $a$. (You should find such a sequence)
Then it is easy to see that although $(-t_n \ ,t_n)$ is in your topology for each $n \in \mathbb N$, the union
$$\bigcup_{n=1}^\infty (-t_n \ , t_n )$$ equals to $(-a,a)$ which, clearly, is not in your topology.
So, actually the only thing you have to do is to find such sequence $t_n$.
