Dear @jpfeifer, thank you! Might it be related to government spending and fiscal rules?
As is it here:

I tried to work it out with different rules and shut down shocks but still have this issue.
Should I assume government debt (B) to be 0 at steady state? If so, I cannot use taylor type fiscal rules. Could you please suggested any other type of fiscal rules? I run out of ideas :((

STEADY-STATE RESULTS when I use wrong Taylor rule for monetary policy.

Y 1.50429
C 1.05625
CR 1.05625
CNR 1.05625 B 36.6369 B_R 73.2739

Dear @jpfeifer Yes, I have lump sum tax and corresponding lump sum tax rule in the model.
In order to balance the budget constraint of Ricardian household I put a manually calculated: STEADY_STATE(ss_tax)
and for Non-Ricardian: T_NR.
and in the aggregation:
T=lambda*T_NR+(1-lambda)*T_R
Since I have it already, it does not change the explosiveness(

@jpfeifer If I just leave T_R for Ricardian it gives:
Error using print_info (line 32)
One of the eigenvalues is close to 0/0 (the absolute value of numerator and denominator is smaller than 0.0000!
If you believe that the model has a unique solution you can try to reduce the value of qz_zero_threshold.

Dear Prof. @jpfeifer , I really appreciate your help in dynare forum’s questions. I made a simplified version by dropping wage rigidity and replacing public spending rules with more “simpler” form:
ghat=rhoG*ghat(-1)+eps_g;
where
ghat=(G-Gbar)/Ybar;

and tax rules as well.

It works, but can I use it? Many papers that I saw employ Leeper/Taylor type rules though linearized one.

This suggests that the problem is indeed with the fiscal rules. You may go for Leeper/Taylor rules as you call them, but they may need a different parameterization.