1640 is a huge number. For instance, if you consider each torrent to consist in a single byte each (so they are quite uninteresting torrents), and you pack them all on 10 TB hard disks (for a torrent to exist, it must exist on at least one hard disk on the planet), and if each such disk weighs about 100g, then the total weight of the disks is about 24 billions of times the mass of the whole Earth.
So I am quite confident in stating that this number will not be reached.
Risks of collisions are actually higher. If you accumulate 160-bit hash values (outputs of SHA-1) then, on average, the first collisions should appear after about 280 values (this is known as the birthday paradox). So the situation you describe may occur much earlier than after exhaustion of the 1640 space. But 280 is still huge, and cannot be practically achieved (280 torrents mean more than one hundred thousand billions of torrents per human being on Earth).
If a hash collision occurred, then streaming either of the two colliding files would most probably cease to work; downloaders would obtain a mixture of both files. Other torrents would remain totally unaffected.
An interesting question is whether you could, given an existing torrent T, handcraft another T' which hashes to the same value -- it would be used as a weapon to prevent usage of T. To do that, you would have to break second preimage resistance of the hash function. SHA-1 has a few known weaknesses, but for now there is no known method for obtaining a second preimage, except luck (try random values until one matches). Luck (aka "brute force") has average cost 2160, which is completely out of reach of today's (and tomorrow's and next decade's) technology.