The Polish/Belarus border crossing at Brest is particularly sticky due to the influx of stolen vehicles. For many years I have drievn all over the region with very few major problems..but in planning a Belarus trip two years ago, I threw up my hands after investigating the red tape...we train-ed from Minsk to Brest..then hired a driver to take us to my wife' mother's birth village about 40 kms, to the south. There in Domachevo, the Border Patrol gave us a very difficult time even though we had a taxi driver. Held us for about a half hour, questioning what the hell we were doing there, yet we had full official visas in our passport pages...a small town on the border. Take a train, Jerry..believe me. Belarus is the only dictatorship remaining in Europe...say no more.
stu (Ihave pix of the area if you wish to see them)

Check that you can take a car from Italy to former Eastern Europe.

You may not even be allowed to take your rental car into Poland. You need to check with the company you are hiring from what restrictions apply with regard to countries you can visit.

I know Belarus is weird, dictatorship, etc. I expect some degree of problems.
But I am mostly interested will problems be beyond 1-2-3 hour delay. Will I be banned to enter Belarus altogether for whatever reason by either EU or Belarus?

I think it really depends on the rental car company's rules that you rent from. You may find it difficult to find a rental company that will allow this at all.

It is the rental company which will give you problems, not the EU or Belarus, assuming you have the right visas.
If you take the car into Belarus without their consent and knowledge you will be uninsured and in deep deep doo-doo if anything happens.
Many companies will not allow their cars to be taken to Poland either due to the high risk of theft.
You MUST check with the rental company. Be honest with them, and see what they say.

This is from Kemwell - renting a car in Italy, but the country, and the company your rent from, is irrelevant. These restrictions are pretty standard.

Scroll down to Rental Restrictions and restricted countries.

http://www.kemwel.com/rental-car/Italy-driving.cfm

"These restrictions are pretty standard."

They aren't really. By and large, most Western European hire cos are reasonably happy (often, of course, for a fee) about low-value cars being taken into Slovenia or the Czech Republic and downright refuse even to think about expensive cars being taken into Belarus, Ukraine or Moldova. In between, different cos have all sorts of rules - and it's likely to be a lot easier to find companies east of the former Iron Curtain prepared to allow at least low-value cars into Belarus.

If you can't find a co prepared to allow this for an Italian-hired car (and I'd be amazed if you did) you probably will, probably via a specialist Polish site, find someone in Poland, Latvia or Lithuania who is prepared to.

Note, though, that the rules for car documentation, insurance and driving licences will differ from the West, and need to be checked on the AA site (http://www.theaa.com/motoring_advice/overseas/countrybycountry.html?mySelect=%2Fmotoring_advice%2Ftouring_tips%2Fbelarus.pdf&go.x=62&go.y=20)

Also remember that rules (such as absolute zero alcohol and absurd speed limts) will be enforced more obsessively than anywhere in the West, and being found guilty (= being accused) may well invalidate your insurance.

Personally, if I found an Italian hire co prepared to hire out a car for Poland or the Baltics (and even that's not totally straightforward), I'd take it somewhere like Bialystock or Vilnius, put it in my hotel's secure garage and get a train or taxi across the border.

The rules are pretty standard Flanner - you have to tell the hire company if you are planning on taking the car to the countries listed. They will not allow some cars to be taken to anywhere in the former eastern bloc. Some they will allow. But I doubt many will allow them into Belarus, no matter what class of car it is.
If the OP does not inform the company of his plans he could be uninsured and as I said in real trouble if anything happens.

"The rules are pretty standard"

They're not. In 1992, I drove a hired BMW from here into Poland: in 1990 a hired Merc into Czech. In both cases most insurance companies were horrified, and it took me (well, actually, my secretary but cetainly not the hire company) some energy to find an insurance co who'd cover it. We had a relationship with the hire co in which they were relaxed about our doing the negotiation.

I'd guess Hertz Italia wouldn't be: Cheapo Cars Polska Sp. z.o.o. in Bialystock (or their cousins in Lithuania) probably do this, at least for Fiat 500s, every day of the week.

Its really strange that some rental car companies don't let cars to travel to countries like Czech Republic, Estonia and Poland, they are EU members and all rules are supposed to be the same for all EU countries.

What is practically going to happen: will EU border patrol stop me from crossing the border? Will Belarus border patrol stop me?

You might not have the proper insurance papers for that country, and in any case, you will not be covered in case of accident, theft or other damage to your car.

Surely this rental car company doesn't even allow to go to Poland, and they never heard of Belarus, they asked where is that after I spelled it. Very educated of them.

I am very much into this trip. I need to visit relatives in Belarus, and I always wanted to visit Prague, Krakow, Vienna.

Can I just buy insurance on my own? Will EU border officials stop me for not having insurance? Or Belarus officials will stop me for not having insurance? What will happen if I try to go?

When I entered Romania, I saw a series of booths selling insurance. I believe that this was for Romanians wishing to enter Hungary, because Hungary would not let Romanians in without health coverage. Similarly, you could be stopped at any border if you did not have proper insurance and that country's law required that every car be properly insured.

Even if you had your own insurance, keep in mind that you probably would have to front the cost of any repairs, possibly pay the rental agency for lost rental time on the car, and then argue with your own insurance company for reimbursement. And if the car had serious damage, do you think that it could be repaired in Belarus or driven back to an acceptable repair shop, or would it mean paying for towing?

Since you wish to visit big cities (Prague, Krakow, Vienna), where cars are more of an encumbrance than facilitators, why not use public transportation?

I am not going to drive all over the large cities. I will use the car to get there, and to visit little towns in between. I like to travel and to see what countries really are, not what tourists are usually shown in tours.