On a full suspension MTB, does a high seat post (extending from frame by 25cm, 10" or more) contribute to frame weakness, should one look to have a bike with a tall seat tube, or are MTB's designed like this and shouldn't break?
There is usually a minimum insertion marker on the seat post to indicate the minimum amount of the seat post which should remain in the seat tube. The frame manufacturer might also specify a minimum insertion depth for the frame. It is not recommended to run the seat post inserted any less than the greater of these two this as it can put the frame under stress and you risk cracking the frame at the top of the seat tube (I've seen this happen first hand).
Having said that with a 400mm seat post I think you'd be able to have 250mm extending from the frame and the 150mm remaining in the seat tube would be sufficient. You'd have to check the seatpost/frame recommendations but if I remember correctly the Rockshox Reverb dropper posts have an 80mm minimum insertion depth, so 150mm is certainly not extreme.
Mountain bikes in general are going to have a more seat post out compared to a road bike as that is the frame design. Mountain bike have a sloped top tube and road bike flat or slightly sloped.
If you are using the bicycle as designed then it should not be a problem. Don't buy a small frame if you are 6 foot and put a monster seat post on it.
Mountain bike also have bigger diameter seat tubes - typically 31.6 versus 27.2.
As stated by chored there is a minimum insertion distance.
There are also a few rules of the thumb to know how much of the seatpost to leave in the frame. If the limit line is not visible on the seatpost, te safe way is to keep 1/3 of its length in the frame. Although it's more than enough for most models (it's around 10-15cm depending on the length of your seatpost), some more expensive ones tend to have thinner walls towards the bottom, which is why going below minimum is a risk.
Also the seatpost should be inserted so that the bottom of the seatpost is below the top tube of a frame (if a given frame has a physical top tube), so the top tube acta against the loads on the seatpost.