What risks are involved with publishing my data and figures at figshare? What are the disadvantages? Did anyone actively decide against doing so, and if so, why?
Well, some journals have a "no prior publication" rule in place. These rules often preclude publishing anything that has already appeared in online databases that are freely accessible, so publishing figures in advance of a publication in a medium like this could be construed as violating such a policy. Since anything publicly posted gets published under a CC-BY license, so this almost assuredly counts with those journals as "prior publication."
In addition, anything that could be considered "confidential" or "proprietary" information definitely should not be posted on such a site until you have secured the necessary permissions from project partners and affiliates. That would also apply to "private" hosting of the figures.
Your own university or research institution may have policies that have some bearing on the ability to use such sites, so check with them as well. Better safe than sorry.
(Note: Those cautions aside, I do think it's a clever idea, and I hope it can catch on!)