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It seems fairly likely that this would only successfully work for a couple of times.
Violating airspace, particularly on a frequent basis, is a pretty easy way to get shot down, Russia or not. It's one thing for a helicopter to stray a mile over the border and pretend "Oopsie! Our navigational equipment is shit!" and another to fly 40-60 miles into Lithuanian airspace across it (assuming a shortest route starting in Belarusian airspace).
It gets into the game theory of things. How likely is Russia to declare war on NATO over losing a single helicopter which is unambiguously doing something it shouldn't do? A war that to all evidence would result in the complete destruction of their military with near impunity. The answer is simply, not very. Oh they might do something stupid like lob a couple cruise missiles over the border in "revenge" and put the ball back into NATO's court. After all, if they did that, Russia can always shrug and point out that THEY didn't initiate a full invasion/attack that leads to WW3 and an almost certain nuclear exchange, so is NATO going to? NATO will likely in this situation respond by basing more equipment in Lithuania (to shoot down such missiles) and increase sanctions on Russia.
The end result there being, Russia lost a helicopter, Lithuania lost a few buildings (admittedly, potentially large apartment complexes) but ultimately the blockade remains and further escalation is avoided. Neither side "feels" like it came out weak in the exchange, and neither side has committed themselves to WW3.
However, it's unlikely to get to this point, because Russia can (at least during the warm months) ship supplies from St Petersburg through the Gulf of Finland to Kaliningrad. I can't find any specific information indicating that Finland or other nations legally can close the Gulf to the transit of Russian vessels in the same way that Turkey was able to close off transiting to/from the Black Sea. Admittedly, that was with approximately 2 minutes of Googling, so someone who knows better might say otherwise. The winter months probably put the kibosh on this plan though, but I'm unfamiliar with how much freezing actually occurs.
Regardless, the likely result is that through one method or another, this situation gets punted a few months down the road.
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