Is boiling tap water sufficient?

Water can safely be drunk after boiling it if the contamination is microbiological. Chemical pollution typically cannot be removed that way.

Since you are mentioning Pridnestrovi: Apparently that is an autonomous region of Moldavia along the river Dniester that is trying to gain independence.

I looked up sources of pollution and other water issues there and found an official report of the environmental ministry of Moldavia. On page 33 it states that

the quality of groundwater abstracted for drinking purposes is particularly poor
in the central and the southern region of the country, where half of the groundwater
samples do not comply with quality standards for chemical parameters. Microbiological
pollution is also widespread […].

The natural and man-induced pollutants mentioned include "nitrates, pesticides, sulfates, etc." against which boiling will not help. Some of the substances like nitrates are more harmful to infants than adults, and pesticides and arsenic accumulate over time so that ingesting small amounts may not be acutely harmful: Drinking water requirements are making sure that drinking large amounts daily over long periods of time is not harmful.

But the standards are there for a reason, and the bottom line is: The pollution is likely not only biological but also chemical, with an unknown mix and concentration of of pollutants. Don’t drink it. Boil or otherwise disinfect it it even for teeth brushing or personal hygiene. Probably try not to shower, or keep it short and ventilate the bathroom.

"Don’t drink from the hot water tap" is universal advice

You probably know sugar dissolves a lot easier in hot water than cold. That’s because hot water really wants to absorb minerals a whole lot more than cold water.

Which means, hot water leaches chemicals from pipes much, much worse than cold water. We try to use low-leach materials for pipes, like PEX, but the amount is never zero. the point is, it’s much worse with hot.

It’s not just the hot water transiting through the pipes, it’s also the hot water sitting in the tank. At the least, most tanks have a zinc "sacrificial anode" to stop corrosion of the tank proper, so that zinc oxide is getting in the hot water for sure.

So, the clever person thinks "If I use from the hot water tap, I’m that much closer to boiling and it’ll save energy/time". That doesn’t actually save energy (because hot water isn’t free), and the saved time isn’t worth the added leachate.

So water you are boiling for consumption should be drawn from the cold, always.

Further, the water which has been sitting stopped in the pipes has had time to leach. You are better off running the water flow for as long as it takes to remove the cylinder of water which has been standing in the pipe for hours. Unless you’re in the American West…

Separate from that, hot water at the wrong temp breeds bacteria

such as legionella. This was something discovered in fairly recent science, particularly during the Flint water crisis. As such, it’s currently considered best-practice to hold tanked hot water at 60C/140F.

This in turn requires anti-scald mixing valves on all water outlets – fortunately most modern "joystick / 2-axis" style faucets include those.

Needless to say, if 60C kills the bacteria in hot water, 100C will kill the bacteria in cold water.

Boiling is not guaranteed to be sufficient for a couple of reasons. The big one is that simple boiling does not remove many chemical contaminants from the water (it may cause some complex ones to break down, but some really nasty stuff, such as botulinum toxin or assorted lead salts, will survive it just fine). If the water is suspect, a combination of active filtration and distillation will get you the cleanest water possible with ‘common’ tools (any cleaner requires chemical treatment, which has it’s own problems, or industrial scale desalination and distillation). However, if you’ve got actual municipal water to work with, simple boiling is usually sufficient.

As far as the hot water, there are two issues there. The first is that it’s only lukewarm. Additionally though, the water heating system is probably tank-based, which means you have a large amount of water in the dark sitting around at that temperature for potentially a very long time. Combined, that means that the hot water system is a breeding ground for microorganisms, which means it’s not exactly safe to drink without boiling. Those circumstances though lead to some risk of microorganisms that naturally produce toxins to grow, and as mentioned above boiling will usually not completely eliminate those toxins.

All together, given that you probably don’t have filtration and distillation equipment available, your safest option is to do as they suggest and use only the cold water supply, boiled, for drinking and cooking.

As an addendum to the existing answers…

No matter where in the world you are, or whether the water is considered potable by the very highest standards… never drink from the hot tap, whether you boil it or not.

The hot supply could be tanked. Standards for header tanks vary from unreliable to none. Frequently, the tank will be in an attic or roof space, often open-topped & at the mercy of every insect, bird, rodent etc that can get to it or die & fall in it. This will often be accompanied by an inch of unidentifiable sludge in the bottom of the tank.

The only time you could even think to drink from the hot supply is if you know for absolute certain is it on a direct ‘instant’ heater, straight from the regular cold supply.

As Willeke already pointed out in her comment, it depends on why the tap water is not safe to drink.

Boiling the tap water will only kill most of the bacteria and viruses in the water. They are probably not common in tap water, but there are also so called thermophile bacteria and viruses, which can easily survive the temperature of boiling water, some species even start to thrive well above the boiling point of water.

Poor tap water may however not only contain bacteria or viruses which can be killed by boiling the water, but also all other kind of dissolved filth, sewer, agricultural (fertilizer, manure) or industrial waste.

You can not see whether water needs boiling, filtering or other ways of cleaning.

Clear water can have a chemical contamination which you cannot get out with ‘in house’ cleaning methods.
Lead pipes do add tiny specks of lead which you also cannot see, or not see with your bare eyes.

While things you can see may not be nasty, just visible. Like many kind of algae.

So check with the people who told you that you can not use the hot water supply, and have to boil the cold water supply, and do what they tell you or buy bottled water in the mean time.

It can be that the hot water supply is no worse than the cold water supply, but it can also have additional problems. And until you know what those problems are, you can not know what you need to do to make that water drinkable.