To move or rotate any object, we need some force, this force can interact with object in different ways, contact force(electromagnetic force between atoms), magnetic force etc. All these forces have interaction particles.
It seems that force that make gyroscope precession is non-contact, not existing force, doesnt make sense that object can change location in space without any "interaction particles".
What are "interaction particles" that cause gyroscope precession?
In the process of gyroscopic precession:
For the gyro wheel there is continuous internal relocation of (angular) momentum. That is to say, there are continuous bending forces as some areas of the gyro wheel are being accelerated and others are being decelerated.
There is a case that also has internal relocation of (angular) momentum, but easier to understand. It's a case that is commonly referred to as 'Feynman's wobbling plate'.
When a spinning plate is thrown up in the air the angular momentum of the plate will be constant; constant both in magnitude and direction.
Any wobble of that spinning plate will not go away when the plate is free from the throwing hand; if that plate would be in a vacuum (hence no air friction) then the wobbling will sustain forever. Youtube video by Jacob Forster: Feynman's wobbling plate
The fact that the plate is rigid is an essential element. It is because of that rigidity that the bending forces do not give rise to dissipation of energy. Comparison: making pizza dough stretch by spinning it (giving the dough freedom to stretch by throwing it upward). You never see that spinning pizza dough wobble. That's because the pizza dough is not rigid; any bending force immediately dissipates.
In the case of a gyroscope:
A necessary condition for gyroscopic precession to occur is that the gyro wheel is rigid. It has to be rigid because in order to display gyroscopic precession the gyro wheel has to support internal relocation of momentum.
There is a 2012 answer by me in which I describe the mechanism of gyroscopic precession. That description explains why it is that when a spinning gyroscope is released it doesn't flip down, but instead settles to gyroscopic precession.
In that description I do not use the abstract concept of spin angular momentum. Instead the explanation capitalizes on symmetry.
Photons and gravitons, not that that provides any meaningful physics insight.
The question is somewhat misguided. For instance:
"To move or rotate any object, we need some force"
Do we? Newton says, "An object in motion stays in motion unless acted upon by an external force". So here we have a moving object with no forces. I feel the same way about rotating.
Re: Exchange Particles. I know popular science communicators love talking about forces being caused by exchanging particle, and show images of (for example) two canoeists throwing a ball and the canoes move away from each other--this always seems to cause more confusion than clarity. People ask, "but how can a force be attractive?" without considering virtual particles can have negative energy...it's all far too literal.
Virtual particles are lines in a diagram representing a term in a perturbation series expansion of an amplitude, that is, there is a force field that mediates the interaction and when we calculate that interaction, it has a coherent series expansion that looks like individual particles. I emphasized "coherent" because it means the intermediate field state isn't "one virtual photon" or "2 photons, one which turns in an $e^+e^-$ pair", and so on. As with the Young's Double Slit Experiment, where the particle goes through both slits, the intermediate field state is all the possible intermediate states, which looks like: a big messy field.
So we tell laymen it's just simple "exchange of force carrying particles", but those are just approximations to a big messy field, so maybe it's better to skip to the ending: forces are carried by big messy field, which is what we all thought in the first place.
And there are 4 fields to choose from: the electromagnetic field, the 2 nuclear forces, and a classical field: gravity.
So EM and gravity make the gyro work, but you already knew that.
The pair of forces that cause precession in a gyroscope or a spinning top are the normal force at the pivot and gravity, which together create a couple on the gyroscope. I don't know what you would call the "interaction particle" for gravity, but I don't see anything mysterious about precession.
