"Please drink more wine."
Translation:Bitte trinkt mehr Wein.
'Trinke' (and 'Trink' too ) is the imperative form for the singular second person while 'Trinkt' is used for the plural second person. So 'Bitte trinke mehr' is used addressing one person while you'd use 'Bitte trinkt mehr' to address a group of people. But, of course, if you talking to a person or a group of people you'd normally address with 'Sie' you'd say 'Bitte trinken Sie mehr.
How about using the infinitive form as a directive? Bitte mehr Wein trinken. Could this be used on a sign, spoken, or is it altogether wrong? I have seen this grammatical construction used before, although I accept that it is not grammatically one of the imperative forms.
it's gramatically correct, but "trink" sounds more natural and is more widely used. I personally would never say "trinke".
I typed "Bitte mehr Wein trink" and was dinged wrong. Is it because of the missing "t" at the end of trink or the word order that's the problem (or both)?
The word order is wrong. Using those words you could write Bitte trink mehr Wein and it would be correct. The -t at the end of the verb is just used if you're speaking to more than one person. Without it, it only makes sense if you're speaking to one person. So, Duolingo will accept both forms.