Is one or both of these sentences grammatically correct?
"You have allowed us to REMAIN OPTIMISTIC"
"You have allowed us to CONTINUE OPTIMISTIC"

The 2nd sentence sounds awkward... but is it wrong? If so, why?

Only the first sentence is correct. The word "optimistic" is an
adjective (see Webster's or any good dictionary) and properly modifies
"us" in the first sentence. The proper word in the second sentence
would be "optimistically". This is an adverb, and properly modifies
the verb "continue", describing how you are continuing.

Or, the second sentence could be: "You have allowed us to continue to
be optimistic."     No charge.

Thank you! I see what you are saying...

But, is it possible that in both sentences the adjective "optimistic"
modifies the subject "us" ?

If "continue" and "remain" are synonyms, why would this not be proper?

In that same way, would it be proper to say "you have allowed us to
remain to be optimisic?" It sounds really awkward but is it wrong?

Remain.

State of being/linking verbs do not show action; they just show that
something exists.

Example:

The chairman remains confused about how to vote on these policies. 

Nothing is happening either physically or intellectually. REMAINS
simply links HE with his state of being: CONFUSED.

http://grammar.uoregon.edu/verbs/linking.html

"Continue" implies some purposefully action is taking place.

The evaluation of the second sentence is less a question of grammar
than of the usage of the word "continue."  To say "We continue
optimistic" is uncommon usage, but not unheard of.  It seems to be
particularly common in writing about business.  For example:

"Three in four CEOs (76 percent) continue optimistic about the
economy?s prospects over the next 12 months..."
http://www.barometersurveys.com/production/barsurv.nsf/vwAllNewsByDocID/8382C3BAAC30A0D585256EEF0074E0CE

"Industry spokespersons continue optimistic, however."
http://www.balancingact-africa.com/news/back/balancing-act_116.html

(I realize that Internet specimens are not proof of correctness.)

A comparable construction like "The weather continues cold" sounds a
bit less weird to me.  A Google search finds hundreds of occurrences
of phrases like this just about weather.

Some dictionaries, such as the Merriam-Webster Online Dictionary, give
"stay" as a synonym for one of the meanings of "continue."  Certainly
"we stay optimistic" seems acceptable.

http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=continue

So my vote is that both sentences are acceptable as standard English.

I agree entirely with Seakinginfo and Voila.
The examples posted by Efn-ga may be influenced by newspaper
headlines, which often leave out words that the reader can infer. 
Such usage slips into texts and may become standard.

Just to confuse the issue:  Two parallel constructions that are
correct but have different meanings.

"That remains to be seen."    (We still have to wait and see if it happens.)
"That continues to be seen."  (We can still see it.)