Not that much compared to the volcano itself. There would of course be radioactive fallout to worry about (which would not normally be present in a volcanic eruption) but the damage would not be on the "Hollywood Thriller" scale. But if the volcano is already active enough to have magma where you could reach it and do something to it with a nuclear bomb, that volcano is probably already letting out a lot more energy than the bomb.
Mount Saint Helens was estimated to have released the equivalent of a 350 Megaton nuclear bomb -- since the very largest city-killer bombs ever put into production were about 20 megatons (most are a megaton or less in yield), they are basically pop-guns compared to a volcano.
Now, if you could find a volcano like MSH which was already under great pressure, and use the nuke from the outside of the volcano to crack the lava dome to trigger an explosive eruption when you wanted it, that might be pretty impressive. But again -- consider how much power was built up in MSH before it cracked its own lava dome, compared to how much force the nuke could generate to "help along" this process. I think the volcano would have to be right on the verge of blowing in the first place for such a plan to have any hope of success, and there just aren't many such opportunities in any one lifespan.
Submerging the bomb into the magma basically won't work. Magma is typically about 2400 degrees F or hotter, which is above the melting point of almost all metals and certainly above the point at which they lose all structural strength. In addition, nuclear weapons are triggered by the detonation of small amounts of conventional explosives timed to go off in very precisely-controlled patterns, and the heat of the magma would doubtless either set some of these off prematurely or cause them to not detonate properly. Nuclear weapons are designed to NOT go "boom" if anything at all goes wrong in the detonation sequence (this is done as a safety feature, so that for example a warhead won't undergo a nuclear explosion in it's own missile silo if the rocket fuel catches fire). I suppose if you built one heck of a heat shield from the stuff that spacecraft use for re-entry you MIGHT have something that would last just long enough to sink a hundred feet or so into the magma. But to do so you would have to drill through the lava dome or the side of the volcano in the first place, which would kind of negate the point of sticking a nuke in it once you were done -- it would already be erupting by then.
Your one chance of doing something really impressive I think would be if you could submerge a nuke into something like the volcano on Hawaii that has been flowing slow but steady for decades now, and blast open the internal paths enough to get the lava geysering again like it did back in the 1950's. But it is unlikely that there is enough pressure under such a volcano to make this work -- if there was, it would probably already have been going like mad just from its own forces.
As to triggering seismic activity -- if properly placed into a fault line that was already under considerable stress... yes, it can trigger an earthquake and this has happened (see the references below). But to intentionally set of a major earthquake you would probably be far better off injecting water under high pressure into the fault, causing it to "slip" -- this is what causes most natural earthquakes, a fault under increasing tension that suddenly slips because the force becomes greater than the shear strength of the materials. Nuclear explosions underground tend to shatter the rocks and everything, producing a very localized stress-relieved area -- the stressed fault connecting to this area will retract somewhat to back-fill this shattered area but the earthquake probably will not radiate far from the point of the explosion compared to what would happen in a "natural" earthquake. Again, to trigger a significant earthquake you would have to detonate your nuke in a fault that was already on the verge of letting go anyway (otherwise, the rest of the fault away from the explosion site won't have enough stress in it to move anyway so will just sit there). I think this was looked at in the 1960's as a potential "peaceful" use of nuclear power, to trigger regular expected motion along fault lines in order to prevent unexpected larger earthquakes, and the idea was rejected by geologists as being entirely unpractical.
The earthquake that recently devastated the Indian Ocean area by causing massive tsunamis had an equivalent power of about 475 megatons -- so again, as you can see, the power of a natural phenomenon is FAR, FAR greater than anything man can produce, at least so far.
Mount Saint Helens nuclear equivalency:
Indian Ocean earthquake nuclear equivalency:
Earthquake actually caused by a nuclear detonation:
Largest nuclear explosion ever generated, a 50 Megatons test:
Yields of various nuclear warheads -- note that the largest ever built by the US was estimated (but never tested) at 25 Megatons, the largest ever actually tested was 15 Megatons, the largest still in existence in the US stockpile is a 9 Megaton model no longer in active service but retained only for analysis, and the largest still in active service is just over 1 Megaton: