What to do, Furnace do not stop heating when thermostat get off? Furnace start work at preset temperature. When heating cycle is completed, thermostat get off, but furnace is running. When I hit outside in the furnace - it will click and after cooling period - furnace will stop. I think the relay or sequencer is wrong. Do i suppose to replace them both?

That's how it should work .Your furnace will cycle on and off to reach set point .
You don't have to troubleshoot the thermostat. The reason why your furnace cycle is once the heat exchanger get to a certain temperature the board will shut it off, fan would still work to cool heat exchanger down then once cool down furnace will fire up again .

Some thermostats allow you to change the differential value - that is how far below the heating (or above the cooling) set point before calling for heat (or cooling in the case of air conditioning). The built in 1 degree differential setting provides the most even temperature in the space at the expense of more frequent furnace firing and stopping. Many people prefer this mode as opposed to allowing wider swings in temperature for less frequent furnace firing and stopping. If you wish to change this setting, you'll need to replace the thermostat as it is not user adjustable in this model.

On properly sized furnaces there are a few controls to look at. Check the settings of the fan switch. If these are unchanged then there may be a reason the furnace is shutting off due to poor air flow and resulting high temperature. Check filter and verify air flow is good. It may be possible that a safety switch is failing and intermittently on and off. Are all the covers securely in place and hitting any safety switches? If not seated properly the cover may loosen when the fan runs and the switch shut off the unit. There is often a flame roll out switch that could have been strained. Lastly, has the thermostat been working long? Thermostats have internal heaters to stop the furnace which is called a heating anticipator. If the thermostat is near a heat source (heat duct, refrigerator, oven, person, hot water pipe), worn out, anticipator is set wrong (older electric thermostats have adjustable settings) or covered (yes, I have seen some covered) then the free room air cannot measure correctly and the thermostat cannot control what it cannot accurately measure. Of all these possibilities I would start with the thermostat and proper furnace air flow, second check the fan/high limit switch and third go component by component through the wiring diagram looking for defects in each component. If available follow the manufacturer's troubleshooting guide.

A thermostat is only a switch, nothing more. It's job is to cut on and off the cooling/heating equipment based on a preset temperature setting. It sends 24-28 volts to a contactor or relay which does the actual closing (turn on) or opening (turn off) the power to the compressor or furnace, and that's all it does. There could be a problem with the other parts of the system. When the space temperature exceeds the preset temperature the thermostat should provide 24-28 volts to the compressor contactor, to start the compressor. The "Auto-on" switch on the thermostat controls the fan only. If there isn't 24-28 volts from the thermostat to the coil of the contactor, the compressor will not operate,

it may be the element or the fuse for it. if this model has the trip switch to stop getting to hot this may be warn out.

Sounds like the anticipator on the thermostat is faulty.
Try replacing the thermostat.

Try cleaning the flame sensor rod with a piece of extra fine steel wool.

Well with a boiler system like you have it will cost you more money to heat your house that way than if you leave it like it is. With a boiler system it takes a lot longer to heat an area than with a furnace. The best thing you can do is find a happy medium and leave it there at all times. If you read the paper work on your new thermostats they have a set procedure which tells the thermostat if you have forced air or boiler system. Do you have baseboard or in-floor heat how many zones do you have.

Sounds like you have a problem upon start up of heat. Your safteys maybe shutting you down, and restting voltage to your curcuit board allows you to have another heating cycle. Honestly you may due justice calling a tech.