見に行かない the pre-masu form(i-form) of 見る + the particle に + the negative form of 行く
(in this case, probably used as a negative interrogative for invitation)
The i form also has some uses on its own, such as:
To express purpose, with に ni: 食べに行きました tabe ni ikimashita: "I went there to eat".
No. I meant 飲みに行きましょうか? with rising intonation sounds like 飲みに行ってあげましょうか? When said with falling intonation, 飲みに行きましょうか is more close to 飲みに行きましょう or "let's go out for a drink."