Is it possible to take DNA from a human being's mouth and put it on an
object ,say the gummed part of an envelope or a stamp or a teacup so
that it can be "discovered" by someone else later?

Being a cynic, I can think of no valid reason why one would want to do
such a thing.  Would you please enlighten us?

I've seen this kind of thing in movies and television shows. A
criminal gets hold of someone's toothbrush or comb or some other
personal item, and transfers the DNA of an innocent man to the crime
scene in order to frame him. I doubt that it happens often, but I
suspect that it is technically possible.

Quite feasible.  DNA is surprisingly robust.
When yoyu think about it, it has to be or else life would never have
caught on.  That double helix thing is topologically ideal for the
role DNA plays.  How else can you keep two molecules intimately
associated without actually chemically bonding them to each other and
so make access to each individual molecule difficult?