Shared calendar not syncing on your iPhone
Start with where the calendar actually lives: a calendar shared from Google syncs through the Google account on your phone, a subscribed ICS feed syncs through iOS's own fetch schedule, and an iCloud share syncs through iCloud. The fix depends on which one you have, and the most common failure is the calendar being attached to an account the phone isn't fully syncing.
Work through these, most likely first
1The calendar belongs to an account that isn't syncing calendars
Settings > Apps > Calendar > Accounts (iOS 18+; older: Settings > Calendar > Accounts). Tap the account the calendar came from and confirm Calendars is toggled on. A Gmail account added for mail only will not sync its calendars.
2Google-shared calendars need the sync checkbox on Google's side
Calendars other people shared with you don't sync to iPhone by default. On a browser, open Google's calendar sync settings page and tick the shared calendar; it appears on the phone at the next sync.
3Fetch schedule set to Manual
Subscribed calendars follow Settings > Apps > Calendar > Accounts > Fetch New Data. If it's Manual, the feed only refreshes when you open the app and pull. Set fetch to Automatically or an interval.
4The calendar is synced but hidden
Open the Calendar app, tap Calendars at the bottom, and check the calendar is ticked. Unticked calendars keep syncing invisibly.
5The subscription captured a one-time snapshot
If someone sent an .ics file (not a URL), you imported events once and nothing will ever update. Only webcal/https feed URLs stay live. Ask for the feed link and re-add it.
Rarer, but worth knowing
Low Power Mode and background refresh. Extended Low Power Mode defers background fetches; feeds catch up when you charge.
Old subscribed feeds after a password change. Password-protected feeds keep the credentials from when you subscribed. If those changed, delete the subscription and re-add it.
TangoHive calendars ship both formats: a share page that is always current in any browser, and a webcal feed for the Calendar app, so the phone's sync quirks never leave anyone without the schedule.