I've never heard of a cat blowing its nose into a tissue, but I used to have
a cat with severe allergies and she used to blow her nose every time it got
too stopped up.  She seemed frightened after it happened and I'm guessing
that it felt like she was choking.  I don't think it is pleasant or
relieving for a cat to blow its nose.  Their little bodies are not like
ours, so my recommendation would be to check with the vet to find out how to
prevent the runny nose in the first place.

I don't know if this will work but try stimulating
your cats nose with something like a thin blade of
grass or something like that. Just tickle the
inside of his/her nose and see what that does.
Other than that I've never heard of a cat blowing
his nose on command.

Hi Edie,

Cats do not blow their noses.  They sneeze, they share!, they wipe with paw,
they lick paw.  Nose cleaned.

If this runny/wet nose is enough to trouble you, consider that it may also
trouble your cat.  Discuss the problem with your vet and see if it can be
resolved.  If your vet isn't duly concerned, but you still are (I'm assuming
this is partly an aesthetic problem), try training your cat to lick or wipe
its nose.  Put a dab of butter on kitty's nose and say "here's a little
treat to clean your nose".  Kitty will immediately (unless it's perverse!)
attempt to clean its nose.   Praise kitty well when nose is clean, and each
time you repeat this training programme.   Eventually, you may be able to
withdraw the (fattening) butter and simply say "clean your nose".    .....
but DO check with your vet first.

Training your cat to "clean up" is quite feasible.  I used to have a lovely
gentleman feline who, very occasionally, would be slightly dirty under his
tail.  I simply used to check under his tail and, if it was soiled, say "you
need to wash your tail".  Washing commenced!  We continued with this
throughout the 22 years of his adorable life.

Hope this helps.
Spider