When tectonic plates collide, the crust can become thicker at the edge of collision by the folding and faulting of crustal rocks. Because crust has a lower density than the asthenosphere and mantle, the region of thicker crust can rise due to buoyancy forces until it reaches isostatic equilibrium. This model of orogeny is referred to in geology as isostasy.
Therefore the "force acting to increase the height of a mountain" you wish to estimate is the buoyant force on the mountain/root system. And it will go to zero approaching equilibrium. Use dimensional analysis to write down an equation for static equilibrium:
[mountain weight force]-[buoyant force] = 0
m(h+H)Ag - MHAg = 0
Where m and M are the mass density of the crust and mantle respectively. Where h and H are the mountain's height and the mountain-root's depth respectively. The horizontal cross-section area of the mountain is A. The gravitational acceleration is g.
Think about this and you'll see how it may explain how erosion of mountain-tops has caused the uplift of the Cascades, whereas eruption of seafloor basalts has led to the sinking of the Hawaiian islands.