Some fairly recent research has shown that dry hopping affects beer bitterness in a number of different, interesting and even unexpected ways.
To address the main question of whether or not dry hopping adds bitterness: it absolutely can (but not necessarily).
Humulinones:
Hops, and especially hop pellets, contain humulinones, a recently-discovered oxidation by-product of hop alpha-acids. They are suggested to be ~65% as bitter as iso-alpha-acids (i.e. 1 ppm of humulinone would be ~0.65 IBUs), and are also readily dissolved in beer without boiling. That is to say, they can be directly extracted into finished beer when dry-hopping.
This study tested commercially available IPAs and found humulinone concentrations as high as 24 ppm. It also suggests that humulinones are not derived from hops during boiling (meaning their contribution is basically all due to dry hopping). So, given the relative bitterness of the humulinones, it seems fair to conclude that a beer with 24 ppm of humulinone has ~16 IBUs worth of bitterness from dry hopping alone. Not at all a trivial number (though not huge, to be fair).
Hulupones:
Another study (which also confirms some points about humulinone) suggests that hulupones (an oxidation by-product of hop beta-acids) can contribute directly to beer bitterness through dry hopping, though it's much less clear to what degree. Hulupone is shown to be ~85% as bitter as iso-alpha-acid (i.e. 1 ppm of hulupone would give ~0.85 IBUs).
Previous studies (cited in the paper) suggest hulupone levels in hops anywhere between 0-3% by weight, and the study itself doses hulupone extract into beer at levels up to 40 ppm (indicating it is clearly soluble). Though no conclusive evidence is shown (and I could not find any additionally), if hops with high enough hulupone concentrations are used for dry hopping, they should be able to contribute non-negligible amounts of bitterness directly.
Dry hopping and pH:
In the first MBAA study linked to above, it was shown that dry hopping seems to linearly increase the pH of the beer, and that the increase in pH seems to cause an increase in the perception of bitterness.
Dry hopping can actually decrease bitterness:
Interestingly, dry hopping can remove iso-alpha-acids from finished beer (presumably by the vegetative matter adsorbing the hydrophobic acids and falling out of suspension). In yet another study, beers of relatively high bitterness were dry hopped with Cascade hops at 1 lb. per barrel (a modest amount by modern standards...) and were shown to have lost up to 37% of their iso-alpha-acids in the process (51 ppm before dry hopping and 32 after, in the paper).
Some general take-aways:
- While dry hopping absolutely can add bitterness, it can also take it away.
- In general, beers low in bitterness to begin with may have significant bitterness contributions from dry hopping
- More bitter (to begin with) beers may tend to lose bitterness in the dry hopping process, even if some of it is replaced with bitterness from humulinones or hulupones.
- It's possible (and seems feasible) that high-alpha hops, having more of the precursors to humulinones, could contribute more bitterness during dry hopping that low-alpha hops. This appears to have not been studied.
Further, here are two blog posts which touch on this subject in a more homebrew-friendly fashion, including recipes, home-scale trials and discussions with the author of two of the studies I've linked:
I'll also note that the accepted answer's points on polyphenols contributing to bitterness are still spot-on and entirely relevant.