I've gone to the BI and obtained my ACR I-Card, but I've subsequently moved. Do I just need to go back to the BI in order to register my change of address? Am I required to do so?
Yes, you're required to notify the BI of any address changes, or you might be liable for administrative fines.
You need to bring some documents with you, which you'll need to do a little leg work in order to produce.
You need to obtain a certificate of residence from your Barangay Captain, who will want a copy of your contract of sale or lease. Ask your property manager / land lord who that is if unsure.
A photocopy of your passport (bio data page, visa implementation page and the page with your stamp for your latest arrival to the country). They do have copiers on site, but you face lines, copiers breaking down, etc. Just get your copies ahead of time.
Make sure you have this, or you'll have fought your way through traffic in vein.
From there, take your paperwork, your original ACR I-card and your passport to the Main (not the smaller satellites) Immigration office, and ask for the renewal/re-issuance form. Complete that, have it examined along with your other paperwork to make sure it's correct, and you'll be sent to the cashier to pay the fees.
Once you get the receipt, make a photocopy of it, then give them the original, along with all of your paperwork and your original card - and your change will be processed, your new card will usually be ready in 10 or so days (they give you a number to call).
Get there early, around 8:30 AM, and this won't take long. If you're going to Intramuros, just reserve the day - you'll want to explore this incredibly old and interesting part of the city once you're done.