What will happen to the density of an object when it's mass increases?

"Increase its mass" usually means just get more of it, and its size increases too. 
When that happens, the density doesn't change.

But if you have some way to increase its mass but not let it get any bigger,
then its density would increase.

An example of that might be:  Take a jar or tank of air, and pump more air
into it.  You added more mass of air, but the size remained the same.  Then
the density of the gas in the tank increased. 

The density will decrease...its pretty logical