Is it correct to mix tripod and handheld footage? I want to shot some scenes using a tripod, and some others handheld (with an IS lens).

Can I mix both types of scenes in a short (4 minutes) movie, or is it unprofessional?

From what I've seen:


Many movies are shot handheld: it is barely noticeable, but in most scenes when it looks like the camera is on a tripod (for example a 10 seconds scene of a woman who talks to her husband in a restaurant), the background still moves a little all the time.
Short movies where tripod is used use it consistently: all scenes are shot with either a tripod or a slider. An example: Philip Bloom's Postcard series.
You're asking for an opinion, because there's no rule book to consult here. So, IMO...

'Pro' movies are shot with all manner and mixtures of techniques, including dolly, mount, rig, crane, Steadicam, fixed, zoom, drone etc. I think many people would find a short film that mixed tripod with deliberate "shaky cam" to be a bit jarring without a good cinematic justification, but just mixing IS handheld with tripod should be no sin. Story trumps technique anyway -- if your viewers are watching how the camera moves, your story isn't compelling -- or they're film students. (-: