давай(те) пить чай / let's have some tea
давай(те) сыграем / let's play
I'm confused; they both mean 'let's', but why does one take an infinitive and the other a future form?
You can think of both forms as 'future tense':
давайте (будем) пить чай
or
давайте попьём чаю
The difference is that imperfective verbs take the auxiliary verb будем (-ет, ут) to form future tense (будем + infinitive). In this particular expression будем can be omitted without any change in sense: it's either way a suggestion for future action.
So, both Russian versions follow the same template with future.
Давайте пойдем, давайте сыграем, давайте сделаем, давайте купим, давайте скажем - all those Perfect forms denote actions as such, which we propose to start, perform and finish.
Давайте идти быстрее, давайте говорить громче, давайте пить чай - these infinitive forms express processes.
Давайте покупать чаще, давайте говорить друг другу комплименты, давайте понимать друг друга - in these cases the actions are either repeated or not possible to be accomplished.
A good exercise for the third case is very famous among the Russian native speakers. :) Here you are. http://spoemdruzya.ru/pesni-iz-filma/klyuch-bez-prava-peredachi/1037-davajte-vosklitsat-davajte-govorit-drug-drugu-komplimenty.html Btw, a more regular action for the compliments is another, and that's делать комплименты. "Говорить комплименты" is not quite a current norm.
You can use either form, the difference is whether an action is continuous or finite.
давай(те) выпьем чаю / let's have some tea
давай(те) играть / let's play