Phrase meaning exceptionally busy phone lines

A standard idiom is that the phones are ringing off the hook.

North American (Of a telephone) be constantly ringing due to a large number of incoming calls:

once the word was out that we had tickets, the phone was ringing off the hook

ODO

It certainly isn't restricted to North America; it's used in Britain. It's an idiom because it's technically impossible for it to happen: the cradle is a physical switch which disconnects the bell circuit.

Jammed

The standard English language phrase would be the switchboard was jammed dating back when all phone calls were routed by an operator. When too many people wanted to place a call they either could not get through or were required to wait longer than normal.

Once direct dialling was invented businesses (mainly media - newspapers & radio and then television) had operators answering phone lines. Unusually high call volumes led to those switchboard being jammed.

In modern times of IVR and call queues where a busy signal is rare, the lines are jammed still works. http://www.oxforddictionaries.com/definition/english/jam

Melted

However if you want to signify heat you could go with the phone lines melted.

Curiously melted does not imply failure or busy signals the way jammed does. Melted means the phones rang red hot.

We need to have the phone lines melted this week. We need people to melt the phone lines. Not to the House members, you don't need to call them, and you don't need to call the Republican Senators. It's the Democrat Senators that we need to melt their phone lines. ―Michele Bachmann via izquotes.com