What is the sum of all the Fibonacci numbers from 1 to infinity. Today I believe I had found the sum of all the Fibonacci numbers are from $1$ to infinity, meaning I had found $F$ for the equation
$F = 1+1+2+3+5+8+13+21+\cdots$
I believe the answer is $-3$, however, I don't know if I am correct or not. So I'm asking if someone can show how $F = -3$.

None of these answers adhere to analytic continuation, which is clearly what you are looking for. A fairly non-rigorous method for doing this is to use the generating function for the Fibonacci series, namely
$$\frac{1}{1-x-x^2} = 1 + 1x + 2x^2 + 3x^3 + 5x^4 + 8x^5 +\cdots$$
Clearly we get the answer $-1$ if we plug the number $1$ into each side. This does not give $-3$.... I am not sure how you would even get that. Again, this is VERY non-rigorous... see this post by Terence Tao for a better foundation into the topic. While it is true the sum diverges in the classical sense, it is often possible to give finite values to sums which have meanings in some contexts (and are simply absurd in others!). For a basic example, look up Cesaro summation, or even Abel summation.
https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ 
EDIT
Please see this link! Apparently a zeta regularization for the Fibonacci series has been done, and matches my answer.
