I worry that the finish will either dull the blades more than unfinished stock would or that the finish will accumulate on the knife edges, dulling them and be difficult to remove.
Assuming the surface is clean (no grit etc. embedded in the finish) I think the risk of direct dulling is only theoretical, because although a few finishes are very tough or hard they are the exception rather than the rule. And even very hard finishes like conversion varnishes are often applied so thinly that cutter knives would just go through them like they weren't really there (they're just a few thousandths thick in the majority of cases). And it's not like some woods themselves aren't very hard and/or abrasive1.
The latter is a legitimate concern and it's why I believe it is most common for a finished surface on reclaimed wood to be taken off by other means. These days that appears to be most frequently with either a belt sander2 or a wide-belt sander. Finish building up on the blades is a real issue, just as resin buildup from resinous wood species is, and while to be fair it is pretty easily removed I think this still falls firmly into the prevention is better than cure category.
Just to mention another alternative, old finish can be taken off using a suitable plane with surprisingly little effort3, with the advantages that it's much quieter than sanding and there's little to no dust generated4.
A roughing plane of some sort — a true scrub plane or a similar-sized plane converted to scrub duties, or a traditional roughing jack or fore plane — can make short work of taking off even very thick finish such as many layers of paint since their curved cutting edges scoop under the surface.
Because of this scooping action the plane irons can stay sharp longer than you'd expect even when the surface is dirty because they interact minimally with the finish and are mostly cutting wood. And even when (not if!) a speck of embedded grit nicks the iron it's not a big deal and you can usually ignore it because the plane is not producing the final surface. These irons also don't have to be super super sharp like most people aim for on a smoothing plane, so when it comes time to hone it can be very quick — 30 seconds to a couple of minutes at most (I aim to get my irons back in the plane in under three minutes).
If you need to retain the wood surface under the finish
Have to mention this as occasionally you do want to retain the original surface on older wood, under a clear finish especially but even under paint if the wood is old enough. The colour of the aged or 'patinated' surface has taken decades or longer to develop and taking off even a thinnish layer of wood from the surface of boards can get below this (all the way to fresh wood in some cases) removing some or all of the very thing you wanted the reclaimed wood for in the first place.
So in this case it can be worth chemically stripping the boards, despite the cost and effort involved.
One last option, flip the boards over
The underside of boards often have no finish on them and for some projects you could ignore the finish on the other side and use the boards almost as-is. In addition to the saving in work, you retain as much thickness as possible this way.
1 Many species have included silica, including some softer species like WRC that you wouldn't expect to be especially dulling to edges.
2 When using a belt sander for this purpose if the finish is thick, e.g. multiple coats of paint, it's worth using a very coarse belt you wouldn't normally consider, e.g. 36 grit.
3 It's also #curiouslysatisfying seeing the finish come off in long strips :-)
4 Something to be aware of removing old paint and varnish that may contain lead or other heavy metals, unless your dust collection is practically 100% efficient.