This is not a simple answer.  It depends on what switcher you are using.

If you are using Wirecast or vMix though the Wirecast Remote or vMix Remote software then the limit is 255.

If you are using an ATEM switcher, it depends on the model.

All but the 2/ME support only 5 connections.  Your control surface will count as one leaving only 4 for the tally lights.  The 2/ME supports up to 7 leaving only 6 for tally lights.

For large ATEM installations we recommend either using the Wireless Tally solutions or the Wired Tally Solutions and not the Ethernet Based Tally Solutions.