They can get cap relief via LTIR, but you cannot bank LTIR space. LTIR doesn't make his cap hit disappear, but it allows the team to go above the salary cap limit by the amount of the LTIR'd contract. You only bank space while beneath the cap limit. They still have to be cap compliant for the remainder of the regular season, so if Burakovsky comes back before the start of the playoffs, his injury won't grant them any more cap space to add additional pieces at the deadline (as they need to save space for him to return to the line-up).
I'll lay out an example. Let's say the Caps are currently spending to 95% of the salary cap. That current 5% opening they have they can bank towards having more space at the deadline. If a player with a 7% contract goes on LTIR, they are allowed to spend up to 107% of the salary cap only while that player is on LTIR. However, any cap space beyond the normal 100% is not banked. So if they call up a 3% player from the AHL, they'd effectively have a 98% salary cap team, and can only bank 2% of the space during the days that AHL player is in the line-up (rather than the 5% they were baking earlier). They can still spend up to 107%, but once that 7% player is back in the line-up, they have to be beneath the normal 100% barrier.
I hope that made sense.