off road or on road tyre

I have a second hand trek 6500 mountain bike. I love the bike but hardly
use it off road. The back tyre needs replacing. Could/Should I change it
to a road tyre if I'm hardly going off road. The front tyre is okay
tread wise(off road tyre). I am a bit skint, are there any problems if
I run it with the original off road front tyre and an on road back
tyre?
Cheers for any advice.
I also bought a Dawes Giro road bike. I find it a struggle up some
hills - I live in a hilly area nr. Ilkley. I'm a not very fit forty
something. I guess the only thing I can really do is to get a new
chainset to make it easier, barring getting a whole lot fitter. I
wanted to buy a road bike for commuting but I'm not sure if the giro is
classed as a racer, a tourer, or an audax. I didn't really want a
racer but have never cycled a road bike before so don't know how the
giro would compare on the road with an audax/tourer type bike, and if I
should trade in to make life easier.


--
Skunk