There are several claims here. Let’s unravel them one by one.
Is there a deep, hot biosphere?
Russell has answered this: yes, there is.
Thomas Gold claims that oil is not aging vegetal or animal matter, but rather made continously from bacteria deep in the Earth's mantle (a 'deep hot biosphere')
No, Thomas Gold made no such claim. This is probably just a misunderstanding of his argument. In fact, here is what Gold said in the “Deep, Hot Biosphere” paper:
Such microbial life may account for the presence of biological molecules in all carbonaceous materials in the outer crust
This does not mean that the carbonaceous material is formed by microbial life but rather that microbes from the deep, hot biosphere contaminate the material.
The text goes on:
and the inference that these materials must have derived from biological deposits accumulated at the surface is therefore not necessarily valid.
Thomas Gold was a follower of the theory of an abiogenic petrolium origin – i.e. that petrolium is not derived by biological processes but rather from “materials of primordial origin” (quotation from the same paper) – meaning, from before life existed on Earth.
You actually noted that yourself:
Gold's theory is that the methane is from the Earth's interior, and cooks into oil.
But you seem to think that Gold claimed the methane came from biological processes. There are in fact several proposed origins of abiogenic petroleum but even though bacteria from the deep, hot biosphere do produce methane, I cannot find any claims that this contributes significantly to petroleum production, and Gold himself certainly never made this claim [1, p 89].
… [consequently] we would effectively never have to worry about running out [of petroleum].
Thomas Gold indeed made this claim, albeit for reasons unrelated to the deep, hot biosphere. It’s hard to track down sources since they’re not online [e.g. 2, 3] but it seems that Gold’s assumptions are based on the fact that more petroleum was found in some oil reserves than was predicted.
However, modern research has effectively disproved this: there is no limitless supply of petroleum [1–3].
 Geoffrey P. Glasby, Abiogenic Origin of Hydrocarbons: An Historical Overview , Resource Geology RESOURCE GEOLOGY, vol. 56(1), 2006
 Earth's "limitless" oil and gas reservoirs are a myth, says study, AFP, April 3, 2002
 Vielvoye, Roger, Gold's gas theory, Oil & Gas Journal vol. 30, February 5, 1979