It's a fair question - and unfortunately the answers here circle around but don't quite aim straight at the nail of what I think the OP is after, which is fundamentally about rocket fuel. Because, no matter how many resources you have on Earth, when it comes to the specific question of launching rockets, you pretty much have to do that from Earth-bound resources, and a rocket launch uses up quite a bit. And getting bunches of precious metals doesn't help if you don't have any fuel left because those things ain't fuel!
For example, the SpaceX Falcon Heavy consumes around 411 megagrams (tonnes) of fuel per launch, and once you're past the atmosphere, you can consider whatever part of that is ejected, spent. And this uses some of the most abundant materials - hydrogen and oxygen - which are easily derivable from water via electrolysis, though kerosene also appears to be involved, and when you get to hydrocarbon fuels, then you have all the well-known problems of their limited supply.
So yes, if you use them enough, in theory eventually you will run out of rocket fuels so that our current technology will be unable to launch anything more. However, the trick is that this form of usage is actually very minimal.
The "ideal" space-based infrastructure, at least as I would envision it, would use rockets only for transporting humans off of Earth - spacecraft for interplanetary transit would always be kept in space, where they could use forms of propulsion that would be impractical on Earth (such as electric plasma rockets, fusion rockets, nuclear explosives, etc.) for various reasons but which would be very useful in space for traveling about, and for these fuels, you have virtually unlimited supply. Thus, you're not even necessarily talking launches on the size of a Falcon Heavy once you've got enough stuff set up "out there" that you can, say, mine asteroids and other space resources.
And thus this comes to the other answers' points - the relevant point I'm trying to make here is to call the attention to the inevitable Earth-based input that must still remain for human-to-space transport. And the answer for that depends on what fuels we're talking. If we're talking the liquid hydrogen/oxygen fuel which, by the way, is what most prior missions have used, it's as abundant as the oceans, and that is around $1.38 \times 10^{18}\ \mathrm{Mg}$, which, even if we had as many launches per year as airline flights, about 36 million, at (say) a 200 Mg per-launch cost so around 7.2 billion Mg ($7.2 \times 10^9$) of fuel per year, we're still talking on the order of 190 million years to remove it all.
That said, this interval is actually a bit of a surprise - the Earth in theory has about 1 billion years of habitable time left with doing nothing, and this carries the seeming implication we would be able to strip it bare of ocean (thus destroying the habitability) with rocket launches before that time. The time for hydrocarbon fuel will, of course, be far less than this at least if we're talking only naturally-occurring hydrocarbons and not, say, synthesis from $\mathrm{CO}_2$ and $\mathrm{H_2O}$ in artificial processes powered by ultra-high-density energy sources like nuclear reactions.
Nonetheless, considering how that "kicking shit down the can" is what currently is killing us with climate change, if we can foresee it, then we need to think about it, I say.
Note, of course, this is likely not that soon, because a rocket does burn a sizeable amount of the fuel in-atmosphere, leaving water vapor and/or carbon dioxide exhaust to return (though there's also the issue then of the solar UV flux at the top of the atmosphere photolyzing the water vapor and releasing the hydrogen), so likely there would still be sizeable amounts of water remaining after and so it is more reasonable to suspect we would not exhaust the supply, but I don't have the chops to figure just how much that would or wouldn't be.
In any case, we should, I'd say, probably want a plan in place to get off of using rockets in maybe the next, say, 300 years or so (10 gigaseconds) in favor of things like on-ground launchers that use the mass of the Earth as reaction mass.
