Question: I've seen many people add logotypes and other graphics in the copper layers of PCBs, often also removing the solder mask in these places to expose the bare (or rather plated) copper.

Is there anything electrically, mechanically or chemically that makes this a bad idea?

Answer: 
The answer here is common sense.

If the logo is placed on some previously empty space and its presence does not require any component placement or routing change there is no harm in doing it. If you're already using ENIG, by all means open the solder mask and make the logo shinny. I doubt that the PCB fab house will charge you more for the extra gold used. If there is no plating restrain from the idea of leaving exposed copper - that's never a good thing.

If the logo disrupts a ground that is critical for shielding or for controlled impedance traces, or if the logo presence risks creating an undesired coupling path, resist the temptation to do it. Functionality is way more important!
We do this in our designs so that we can save money by totally avoiding silkscreen printing. The text as such will be small, isolated from all other sections in the circuitry and doesn't create any trouble with EMI EMC. Ofcourse this wouldn't be placed some where close to a impedance matched tracks but rather in a place which is easy to read but without harming any other sensitive sections.