Can I use a 15A breaker for a 10A baseboard heater?
Wire can always (or almost always, I'll bet there is an exception somewhere) be oversized. Using a larger wire than required by code will either make no difference at all or only improve things - e.g., by reducing voltage drop over long distances.

Breakers must NEVER be oversized when used as a protective device. Generally this is pretty clear from the particular device - e.g., a device that requires a 20 Amp circuit must have at least matching size wire (typically that would be 12 AWG) and must have a breaker that is exactly 20 Amp. There are certain exceptions - e.g., a 20 Amp breaker (with at least 12 AWG wire) may be used for 15 Amp receptacles, provided there are at least 2 (e.g., a typical duplex 15 Amp receptacle). But you can't put a 30 Amp breaker on a circuit that is rated for 20 Amps (either because of the device specifications, the wire size or both).

Breakers CAN be oversized when used merely as an on/off switch. This is the case, for example, in some subpanel installations. For example, if you have a 60 Amp breaker in your main panel feeding appropriately sized wire to a subpanel, then the subpanel can have any size breaker (though obviously it should be 60 Amp or larger in this example) as a main breaker because it is not providing protection, it is only used as a switch (e.g., to be able to safely work inside the subpanel without going back to the main panel). If you search around a bit on DIY, you will find many examples of using a large panel with a 100 Amp (or even larger) breaker for a subpanel that is actually fed by a smaller breaker in the main panel.

Now on to your situation:


  220 V baseboard heater with a 10 amp draw, wall mount thermostat. Can I run 12-2 wire with a 15 amp breaker?


The first determining factor is the specifications of the heater. It should have some information about what type of circuit it works with. Barring any specifics, 15 Amp is the typical bottom meter size in the US, and is obviously big enough (and should be safe enough) for a 5A or 10A heater. However, more typically with 200 V loads you will see larger breakers as one of the reasons to use 220 V is that you can get a lot more power than with 110 V. In particular, if you will be connecting multiple baseboard heaters on one circuit then you need to make sure that they are designed so that they can be on the same circuit and protected appropriately. Your circuit (including the breaker) needs to be larger than the continuous load (heaters may be on for long periods of time, so they definitely count for continuous load), I believe by 20%. So a 15A heater, for example, would actually need a 20A circuit (12 AWG wire and 20A breaker).

As far as wire, as already noted in another answer:


You need to see if a neutral is needed (you can't share ground &amp; neutral). If so, then you actually need 12/3, not 12/2.
Even if you are using a 15A circuit now, going up to 12 AWG doesn't cost much for 40 feet and gives you more flexibility (i.e., up to 20A) for the future.