Overthrowing Lukashenko could potentially have dire consequences for Ukraine. It would be a perfect opportunity for Putin for an intervention. Not that he needs it right now, must Russians I know are sick of him, which is something I thought I’d not see as they were once staunch supporters of him. Poverty does that to you.
It's a bit more complicated than that I think. A little while ago, Lukashenko's Belarus was very close to Russia, and there was some talk of Belarus maybe eventually becoming a Russian province. That has completely reversed though: Lukashenko is now rather mistrusting of Putin and Russia. He's not quite steering Belarus towards Europe, but trying harder to steer a middle course. In the meantime, he has started fostering a unique Belarussian national identity, by emphasizing differences and promoting the national language. (Belarussian is not Russian, but was long suppressed.)
Obviously, all that is self-serving, as Lukashenko hopes it will give him a little more credit with his people and help him stay in power of 'his own' country in the face of Russian pressure. But all the same, it means that having Lukashenko in power is not necessarily in Russia's best interest. Neither is a revolution that creates a democratic country though, as that might give the Russian opposition the 'wrong' ideas and inspiration. So from my perspective, Putin might actually like the current unrest best. If he can fan the flames and give it an anti-Russian angle, that would give him the pretext to intervene like he did in Ukraine, to 'save' the Russians in Belarus. As
@Rajma indicated, the rest of Europe and the US have never seemed to care much about Belarus, so the NATO response would probably be rather lacklustre.
Or at least, that's what I've been getting out of articles I've been reading in the past few months. Happy to be corrected.
Edit: Just read about another option. Putin helps Lukashenko stay in power, but on the condition that Lukashenko returns to his previous trajectory, i.e., increased integration with Russia, which might eventually lead to Belarus practically becoming an autonomous province of Russia.