Normally, messengers such as Signal, Threema or WhatsApp continue to work without any problems, even if the SIM is removed after activation. For example, it is possible to install the messenger on one device and receive the activation code via another device with its phone number.
However, regulatory interventions, such as those currently becoming mandatory in India, could change this behavior: In future, messengers there will have to constantly check whether an active, clearly identified (KYC-registered) SIM card is inserted in the device. This means that all messenger usage will be tied to the mobile network. This has far-reaching consequences for various risk groups:
At protests or demonstrations, the SIM constraint prevents secure messenger use via temporary WiFi hotspots, which again makes people completely visible to radio cell evaluations and IMSI catchers. When traveling or on vacation, messengers can suddenly no longer be used outside the home network or without roaming. For journalists and people critical of the regime, mandatory SIM liveness means that it is no longer possible to specifically separate the device and SIM, and sources may become easier to identify. For people who are on the run, being tied to an active national SIM card makes secure communication considerably more difficult because SIMs often cannot be taken with them or are deactivated after a short time.
If you are dependent on privacy, you should check whether your messengers continue to work even without a SIM card inserted. E.g. via WiFi. If this is not the case, you may be able to switch to alternative messengers such as Matrix or DeltaChat, which do not require a phone number.