What effects does opening your town's gate have?

While the gate is open:

• The Happy Home Showcase is blocked off
• The Dream Suite is disabled
• Access to the 'Mayor chair' in Town Hall is blocked
• Blathers won't accept donations or assess fossils
• Saharah and similar NPCs seem to leave or are otherwise unavailable (e.g. Gulliver won't wake up) - note this does not apply to Katrina or probably Redd, who have tents.
• Bees, spiders and scorpions will not appear
• Favours for villagers, such as deliveries, cannot be performed

Along with what's already been said, you cannot give Shrunk food for emotions. If I'm thinking correctly, you also cannot purchase another exhibit spot in the second floor of the museum.