While others have illustrated ways in which you can reason about infinite coin flips, I think I can show more directly why your reasoning is incorrect:
So as we approach an infinite amount of coin flips, the probability gets smaller and smaller, but it shouldn't ever reach zero, so the probability shouldn't be zero, it should be 1/+∞.
But just as the probability never "reaches" zero, so the denominator never "reaches" $+\infty$ (even if you come up with a firm idea of what that symbol means, and how you could reach it). All this tells you is that looking exclusively at what happens for any finite number of coin flips doesn't tell you anything for sure about what happens when you conduct an infinite number of coin flips. The infinite is altogether more exotic and strange than the finite.
Here's an illustrative example: suppose I start with the question "If I flip $N$ coins, what is the probability I get $N$ heads?" and decide I'm interested in a version of this question with $N=\infty$. Either of the following sound like sensible ways forward:
- If I flip infinitely many coins, what is the probability that infinitely many of them are heads?
- If I flip infinitely many coins, what is the probability that all of them are heads?
They're both essentially the result of thinking about $\infty$ instead of $N$. But a moment's thought shows they are completely different questions! In particular, the answer (in the usual probability model, as described by Qiaochu Yuan) is $1$ for the first question and $0$ for the second – they couldn't possibly be more different! (Notice that it is possible, say in the sequence $HTHTHT\dots$ to get $\infty$ for both the number of heads and the number of tails, while it's certainly not possible in the previous question to get both $N$ heads and $N$ tails.)
So you can't just write symbols like $+\infty$ in the place of ordinary finite numbers and expect to get a sensible result. In mathematics, infinity is used in altogether more delicate ways: although there are ways in which you can make it a number, or at least behave in number-like ways, it is certainly not the same sort of number as $2$ or $7$ or $\pi$ or $i$. So don't ask what $1/\infty$ is before first making sure that you know what $\infty$ is, and why you care about the result.
Furthermore, if you think that the probability of an infinite number of coin tosses all coming up heads is not zero, well, you are welcome to go away and toss an infinite number of coins, and when you're done come back and tell us what you got. The fact that you can't do this ought to make you think about what you really mean when you ask a question like "what happens when I toss infinitely many coins" and on what authority can you prononunce one or another answer to that question "right" or "wrong". The mathematical answers to this question, I would argue, are neither right nor wrong but merely convenient: they let you think about things in a simple and uniform way that is always consistent and is correct for every testable situation (and not incorrect for any other situation, since correctness makes no real sense there!)