Whilst I like Mr Tennant very much, I suspect that he was not the one that came up with the Signature Edition figure. That was all BCS, Mr. T had two choices, to sign the plaques or not. The fact that he did says to me that he probably was contractually obliged to do so in some way.
He is in no way obliged to sign anything. In fact, unless the rules have radically changed in the last few years (and I doubt there is an actor's union anywhere that would allow such a thing), then actors can't be contractually forced into giving any autographs, no matter what other deals are in place as far as merchandising or whatever else goes. An actor always retains the ownership of their name and signature, just as they only ever sign over the use of their likeness as depicted in a given role, and even then, if used in actual merchandising (and not just standard promotional material) it has to be specifically contracted for. Now they can, of course, choose to sign stuff for free, or be paid for doing so, as part of promotional events or merchandising, and it is pretty much standard that most actors will sign for free, or for the price of return postage, for fans who write them, but no one is forcing anyone into doing so. Actors traditionally get paid an agreed appearance fee for any promotional work they do (including signings) for this very reason.
While I'm sure it was likely part of his contract that his likeness as The Doctor would be able to be used and replicated by BBC for merchandising purposes basically in perpetuity, there is no way he (or any other actor playing The Doctor) is being forced to sign anything, nor could anyone's signature be legally replicated without their permission to do so. No, BCS would have paid him for his time, and I don't imagine that David Tennant would have been all that hard to make a fair deal with, as he probably understands this stuff better than most actors, from both sides of the fence.
Besides, it's likely a nice chunk of change for a couple of hours easy work, signing a bunch of plaques and posing for a couple of promotional photographs. I mean the signed editions are an extra twenty quid each, for three hundred that's six grand. I wouldn't be at all surprised if the actor in question doing the signing saw half of that as their payment (depending on the actor could even be more, seriously doubt it would be less than a third no matter who it is). That's not bad for something that would be arranged to suit their schedule, and wouldn't impact on anything else, but instead would likely slot into a casual couple of hours when they had the spare time. In fact I'd say it's probably about the closest an actor gets to free money.
Hell, even if most of that money went to securing the actor's time to sign the plaques, BCS still have their profit margin built into the original figure cost anyway, and this is just a way to help sell more figures for BCS, so any slice of that additional six grand they might walk away with at all is pure win from their point of view. Especially combined with the amount of attention that the signed editions help shine on their products to begin with.
Point is, nobody forced Tennant into doing this, no matter what way you cut it. These kind of things just don't work that way.
It is going to be interesting to see what happens when they get to Christopher Eccleston's Doctor, assuming that they do. While the rest of the (still living) Doctors should be easy enough to make a deal with, that one I really could see going either way.
It's also going to be interesting to see what kind of exclusive they go for with William Hartnell's Doctor (and Troughton and Pertwee when they get to them). Will they negotiate with the estate(s) for the right to replicate their signatures, and reproduce them as etchings or autopen, just to try and maintain a certain consistency across the line, or will they come up with some other idea for exclusives for those Doctors? I hope we at the very least get a named plaque for each doctor done in the established style, no matter what direction they decide to take for the Doctors that are sadly no longer with us.