With Tianna Crownguard on a battlefield I control, if my opponent has units on Grove of the God Willow, do they get the trigger from Grove at the beginning of their turn?

Yes, your opponent still receives the trigger from Grove of the God-Willow.

According to RiftJudge FAQ (ID 8447 and 8403) and Core Rule 383.4.d, Tianna Crownguard’s ability only prevents opponents from gaining points. It does not prevent the act of "Holding" a battlefield or the triggering of "Hold Effects" associated with that process.

Explanation:

• Tianna Crownguard: Her ability states, "While I'm at a battlefield, opponents can't gain points." This is specifically restricted to the act of gaining points (adding to their score total).
• Grove of the God-Willow: Its ability ("When you hold here, draw 1.") is a Hold Effect (Rule 383.4.d).
• Interaction: Per the FAQs, Tianna does not interfere with the process of Holding itself, nor does she stop triggered abilities that result from Holding. Your opponent will still perform the Hold action, the "When you hold here" condition will be met, and the draw trigger will be placed on the chain as normal. The only change is that if that Hold action would have resulted in them gaining a victory point, Tianna prevents that point from being added to their total.