I am starting to do product photography but I dont know whether I should keep the uv filter on my lens or not.
If you shoot indoors and use it to protect you sensor from UV (which is mostly pointless anyway) then you can remove it. If you use it to protect your front lens, then it's not terribly useful either, product photography being not too risky for cameras (do you need to wear special equipment during the shoot?).
So, yes, you can remove that filter to use your lens to its full potential.
The point of UV filtering on image quality is quite less than with film since the color filters used on sensors tend to be a bit more specific than film emulsion sensitivities were. To make a difference at all, you'd need UV filters significantly cutting out UV light in the near-visible spectrum. Those will be quite expensive and will mainly have some effect on lenses prone to purple fringing in backlit outdoor scenes.
Another thought more relevant in the digital age would be to protect plastic lenses (organic agents like canada balm are not that common any more) from premature aging. This concerns primarily hard UV, an exclusively outdoor concern.
So with regard to getting a somewhat better initial color balance and possibly help with lenses prone to purple fringing, the near-visible UV is the only actual concern. Which illumination even has UV in it? Incandescent, particularly halogen. Most indoor lighting these days is LED or fluorescent. And of course flash. Flash does have UV components but those are usually reduced using appropriate filters on the flash itself.
So after walking through the options, it's pretty clear that indoors only rather expensive UV filters with good blockage of near-visible UV could conceivably make a difference, and this difference rarely has a chance to make an impact. Fine art with strong pigments requiring incandescent lighting for proper appreciation might be the kind of thing where your chances of better results after white balance might improve but you'll likely get better differentiation by using something like a DSC-F828 with RGBE color matrix (namely four primary color receptors instead of three) instead of RGBG. Note that this increased spectral color resolution has not managed to survive in the market even though it should make it easier to capture colors tricky to white balance.
Now you are not talking fine art photography but product photography. You are not going to work with UV-loaded lighting or strong iridiscent pigments.
Your UV filter will mainly be for lens protection. Since it is not going to buy you any noticeable increase in image quality, there are no tradeoffs in return for any decrease in image quality (waves, stripes, flare, doohickeys). So keeping the filter on registers as "might-as-well" only for the highest quality filters that really manage to do essentially nothing at all in the visible spectrum.
How much are you willing to pay for nothing?