ir plays triple duty, being used to construct the periphrastic future, to indicate motion, and occasionally to construct nosotros/as commands. Only two of these three are mutually exclusive, so we get two potential interpretations with vamos a ir a ver.
The first ir can either set up the future or, thanks to being specifically the form vamos (a shortened form of vayamos), it can also substitute as a command, equivalent to English's “let’s…”.
In either case, the second ir expresses the actual action which is to go.
You might be think that since both irs are followed by a that they are both expressing future and that your main verb is ver, but just because something is ir a [verbo] doesn't mean it's a periphrastic future construction.
Vamos a tomar el metro para poder llegar al cine para ver la película
Vamos a tomar el metro para ir al cine para ver la película.
Vamos a ir en metro al cine para ver la película.
Vamos a ir al cine a ver la película.
Vamos a ir a ver la película
Iremos a ver (future sense)
Vayamos a ver (command sense)
The a is really a para in disguise showing purpose, but not necessarily temporality.