Can I put a 2.3 mountain bike tire on what now has a 2.10 tire. 26 inch
In terms on the rims/wheels, you certainly can.
In terms on the frame and forks, you have to make sure that you have enough clearance for the tyre, plus extra room for the wheel to wobble a bit (wheels/tyres distort during normal use, how much depends on wheels, brakes, frame etc) and room for mud/dirt/ other things.
You also have to consider whether the front derailleur (if you have on) will hit the tyre during use.
I would personally not use a tyre if the clearance on each side is 5mm.
EXTRA:
Wheel wobble link 1 They have issues with tyre rub, possibly due to weak wheel
Wheel wobble link 2 They have issues with tyre rub after increasing tyre width