In team fights you are a frontline hero - you need to be in a position to latch onto an enemy that is overextending to start the fight. Remember that you could start a fight easily. Nonetheless, sometimes you lack the control and damage to finish it, so stick close to your team.
Consider how you use your Phantasm - it is usually best to cast your illusions before initiating or entering the team fight. This could help you kill your target before it even has the opportunity to respond as it will get hit by all illusions when you cast your Rift.
As you have no escape mechanism and it is hard for you to disengage, it is usually the case that when you enter a fight, you either win it or you die. This is why it is wise to pick your fights. Usually (especially in the mid game) you should be potent enough to be in position to win most fights, with some proper help from your team. However, later on, team fighting could get riskier. For example, it could be unwise to enter a fight without having your Phantasm ready.
You are a carry and a tank, so try to stay between the enemy carries and your squishier teammates. Don’t go chasing after supports that are safely positioned behind their carries even though you might have the chance to reach them with Rift. This way you will allow the enemy carries to run into your squishier heroes, and you don’t want that to happen. If a support is overextending, it is a good idea to Rift it and insta-kill it with your illusions. Otherwise, stay in the front lines and fight the enemy carries. With the help of your illusions, you will be able to win most man-fights.
Saving your Phantasm to dodge spells is NOT a good idea later on. If you get the opportunity (a projectile is flying towards you), you should take advantage, but don’t wait for such an opportunity. You want to have the illusions for the whole fight, otherwise, you will be much less useful for the duration without them. Bear in mind that if you get on low HP before you cast your illusions, your illusions will be on low HP when you cast them as well, which means that they will be much more susceptible to AoE. Therefore, it is best to use Phantasm before the fight (before you initiate), or at the beginning of the fight (just as the enemy team is initiating).
In case enemy heroes have powerful AoE spells, you might want to save your ult until those spells are on cooldown. Be careful not to get into a man-fight without your illusions, however- they are your primary source of damage.
Example: