No, it doesn't interfere.
For instance, when Anytone radios send APRS-digital data, it uses a separate talkgroup, 311999, on timeslot 1. You may be using a voice talkgroup that is on timeslot 1 or timeslot 2. The radio switches between these modes seamlessly during transmission. This data transmission doesn't depend on talkgroups and happens between voice activity.
(Note about APRS-analog: Similarly, APRS analog changes frequency and settings automatically between voice transmissions. There's no interference between voice and APRS operations.)