If science can't prove the existence of other dimensions forever

I think that other spiritual worlds or a faith in its existence usually has no essential influence on our way and everyday life. So such beliefs can be considered as interpretations of one’s experience, which can be helpful and appropriate for his practice. Especially if this experience becomes quite unusual. For instance, I expect that deep spiritual practicians of different ways even can see things in which they believe.

... yes and no (no, later on)

It is a part of Right View to believe in karma in general, including the process of reincarnation. Honestly though karma is pretty common sense and proven by science, psychology, and everything. Reincarnation is a step further with the principle of karma, combined with Newton's 3rd Law.

The details of the Thirty-one Planes of Existence are very believable due to its correlations with many other phenomena, I would suggest Professor Bavo Lieven's beautiful, rich, and very motivating cosmological analyses in the first half of his book, "The Mind Experiment" to really get some tasty details on the 31 Planes.

And now comes the "no" part of the answer.

The Buddha himself said many times throughout the suttas that when you attain 4th jhana, you will either spontaneously be able to see all these realms as easily as you see the current Desire realm. Or one will have to develop the paranormal power to do so if one doesn't spontaneously see (how? read the Path of Purification, a step by step scientific experiment to do all this).

It's really just about developing the 1st-4th jhanas (a sign of mental stability and concentration power--not comparable to tripping out on shrooms and seeing such things) and verifying these things for yourself.

So... no. You don't have to take it as belief especially to progress with Buddhism.

You do have to believe in the concept of karma (which isn't hard... try doing an experiment of bad actions and good actions... I did this 15 years ago for a whole year and I certainly felt the results short-term and long-term) to be practicing Buddhism because it is a definition of Right View, the first step of the Buddhist path. From accepting karma as a principle of the universe (Right View)... reincarnation is a short step. And from there the Thirty One Planes are just arbitrary details that can be verified later!

Physics is talking about physical reality. The "planes" of Buddhism aren't (necessarily) physical planes. So you don't (necessarily) have to choose between them. You can believe both, either, or neither.

But "should I believe" implies you have control over what you believe. Is that really true? Can you actually choose to belief any specific thing? Isn't it more the case that you find yourself believing something, depending on what you experience, read, hear, how you act, and so on?

Given that, one approach is simply to get on with life, practicing as you are willing and able, and see what you find. As the Buddha is reported in Maha-parinibbana Sutta to have said, "Strive with earnestness";similarly in Philippians 2:12 of the Christian New Testament, "Work out your salvation with fear and trembling."

Three possible approaches, here.

One, you just acceptt this on faith. Depending on your mindset, this can be something like believing the impossible or groundless, or something more like feeling like you need to a decision. You can't prove tea or coffee is better for breakfast, but you can't stand around all day weighting the alternatives. So you make a decision, aware that the reasoning has some holes.

Two, you look at these as a myth, an instructive story and you don't even put much thought into if the story is true. In our distant past, everything was hearsay. There was no science, no reliable news outlets, and it was just a bunch of people telling stories. People got used to some stories seeming a bit far fetched, but everyone you ever heard in your life was a bunch of plausible sounding stories.

Three, you look at these as patently incompatible with everything we know about the world from multiple reliable sources-- science, journalism, and so on. You either discard that part of Buddhism as nonsense, or alternatively review it and look for what can be salvaged-- for example, some modern readers view the cosmology as a metaphor for states of mind (hell is the state of mind in this world and this body that we put ourselves in when we act unskillfully)