I don't know the relationships between Malone and Booth, yeah they might not like each other just like Luka and Nico did, but they have one goal, that is to make another run in the championship for this year and it seems that they are on track as they are still in the playoffs.
I searched on the net for the reason behind firing coach Malone and i see one of those is an internal conflict between Malone and GM Calvin Booth regarding roster construction as Malone wanted to field veteran players while Booth wanted to focus on young talents but sad to see both were leaving the Nuggets franchise.
That's one of the issues, but that isn't the reason why he got fired. I think it's Malone's way of handling his players.
The conflict between GM Booth and Coach Malone has started way back even the start of this regular season. Coach Malone didn't like the moves of their GM like signing Nnaji to a ridiculous contract which destroyed the whole team if you will ask me, and giving Reggie Jackson a huge contract only to get traded as well. Because of these moves, they can't afford to retain KCP which is a career 40% 3-pt shooter, they also let go of Bruce Brown as well for the same reason.
Why I believe it's the way he's handling players is the reason is because Coach Malone is focusing too much on his starters, and don't give much time to his bench in the playoffs. He's like another version of Coach Thibs, but only for playoffs, and we know that playoff is different compared to regular season. In the playoffs, you could feel the fatigue especially if you played huge minutes, and how many days is the rest? 1-2 days only which for some is not enough. Maybe this way of handling the players is enough for Coach Malone to be fired.
Overall, I just don't want Coach Malone to be fired because of his contributions with the team, but it is what it is. It happened already.