I'll give you my perspective as someone who has abandoned cash games in favor of only doing live MTTs.
I think a lot of this will depend on your current level of experience and your game will change over time. Of course, you need to be sound in things like picking your pre-flop hands, but these are some things that I know I need to work on for my tourneys:
1 - MTTs constantly have you playing with new people, new tables. When I sit down I try to pause and take stock - how many players? Who are the big stacks? Who are the small stacks? Where are they? Get a sense of where you're at in the lineup.
2 - On your way to the final table, you're going to play everything from full tables to short handed. Make sure that you are comfortable "switching gears" between both of those. For example, think about an MTT and there's only 12 of you left. That's two tables, six-handed that then becomes one final table, ten-handed. Can you make the right adjustments for a full table after playing six-handed for a few hours? That's not always an easy thing and I know it's tripped me up in some tourneys.
3 - Be quick on your preflop math. Most people make mistakes when it comes to antes. ALWAYS know how much is in the pot pre-flop. Know what an appropriate half-pot and pot size bet is to open. Don't sit there and watch the dealer shuffle, watch the tourney clock and figure out the ante, the blinds and know what that means for the pre-flop pot.
Hope this helps.