Removing forward lookup zone broke our site - why?

I'm fairly new to the job and trying to get to grips with the infrastructure here. We've moved a site from being locally hosted on our own network to an external host (1&1). I've transferred the DNS hosting from the previous DNS host to 1&1 to keep things simple.

Once everything had gone through, visitors that were external to our network were being directed to the new site on 1&1 but requests from within our network were still going to our own server.

I noticed in the DNS server that there was a forward lookup zone for the site pointing to our own server still. My (admittedly simplistic) understanding was that pausing that zone would then cause the DNS server to get the address for the site from our external DNS servers and our users would start landing on our new site. However, what happened instead was that they were being met with "page not found" type errors. I've resolved it my modifying the forward lookup zone A record to point to the external web server but would like to get an understanding as to why pausing the zone didn't work. Would deleting the zone work? I am reluctant to try that as creating it again will not be as easy as simply pressing "start".

If the zone is not your AD DNS zone then you're free to delete it. As Driftpeasant stated, pausing the zone does not make the DNS server non-authoritative for the zone, it only makes the server refuse queries for that zone. If you're worried about having to recreate it from scratch if needed in the future then set the zone to a standard Primary zone as opposed to being AD integrated. This will then store the zone in a text file (with a .dns extension) in the %systemroot%\system32\dns directory of the DNS server. You can copy this file off of the server before you delete the zone and recreate the zone later from this file if needed.

Is the name of your Windows domain the same as the name of your website? That is, is your Windows domain SITENAME.COM and your website is WWW.SITENAME.COM?

If yes, DO NOT DELETE THE FORWARDING ZONE. It will break Active Directory massively and you'll be in a world of hurt. If you're lucky enough that your Windows domain is SITENAME.LOCAL (or .LAN or whatever other non-publicly-routable domain suffix), then deleting the zone will force external lookups.

Before you do that, though, make sure that you don't have any other weird subdomains on that pointing to internal servers - i.e. TEST.SITENAME.COM points to 192.168.1.2 - then you'll need to maintain the internal DNS zone and just update the A records accordingly.