Let's say humans have finally made a successful colony on Mars, how useful would bitcoin be to transfer value between Earth and Mars?
If you're talking about the Bitcoin blockchain in its current form, it will restrict mining to one of both planets.
Bitcoin's assumptions include a block propagation time between miners is negligible compared to the interblock time (10 minutes). We've already seen the effects of a few seconds block propagation time on Earth: miners being forced to build upon each other's work without validating, to avoid orphaning during that time. This incentive encourages centralization, hurts Bitcoin's security (especially for light clients), and makes mining (even) less permissionless to enter.
The propagation time between Earth and Mars can be up to 401 million km, when they're on opposite sides of the Sun. Following the laws of relativity, it is fundamentally impossible to communicate between them in less than 22 minutes in that case. A propagation time of 22 minutes would utterly destroy any ability to mine on both sides. As there is a delay of 22 minutes before Earth can mine on top of a block created by Mars, or the other way around, but not when a planet mines on top of a block created by itself, you would end up with an almost persistently forked chain between the two. The slower planer would eventually always switch over to that of the faster planet, making the faster planet win far more blocks that proportional to its hashrate. Effectively, the delay between the two planets turns mining into a race rather than a lottery.
And if it won't work with the current implementation, what would need to be changed in order for btc to be used interplanetary?
The above issue could probably be solved by using a chain with a ~1 week interblock time rather than 10 minutes, to compensate for the larger propagation time.
If we're fine with mining being controlled by one of both planets (effectively giving that planet the arbitrarily censor the other planet's transactions), it would probably work fine.
There are other issues though; another problem is partitioning. Bitcoin strongly relies on peers in the network to be able to communicate without ending up on their own islands of connectivity. There would need to be multiple independent communication links between the two planets to avoid a partition between Earth and Mars. If mining is knowingly on one side, the worst effect of a connectivity failure is just the lack of ability to see confirmations on the other side, though it may take hours (=a few roundtrips) to notice this.