Start over.
The instructions for any telephone subscriber who has an alarm system running on a phone line is to unplug the alarm panel cord (not an "RJ45" cord, by the way) from the RJ31X jack. This is a simple test that can be performed without tools or telephone knowledge. In fact, it is a requirement by the FCC that the subscriber be able to do this test.
The above is what we call "theory" or "Telephony for Idealists, Optimists and Utopians."
In the real world, don't be discouraged if you can't find the RJ31X, more commonly called the "series jack." . More often than not, the alarm installer will put the jack INSIDE the LOCKED alarm panel, just so that the subscriber will be forced to call the alarm company for help, incurring a repair fee. Don't panic. The key to the alarm panel is "hidden" on the top edge of the alarm box in 99% of all installations. Failing that, the alarm boxes are locked with only about 6 different keys in the USA. They can all be jiggled open with the end of a paper clip.
If you are lucky enough to find the RJ31X, just unplug the cord and wait a few seconds. Try to get a dial tone at the block or the fax machine.
Just in case the other advice above was a bit confusing, here's another way to understand the wiring: The dial tone MUST, by NFPA standards, first go directly to the alarm panel, via the series jack. It gets there on a pair of wires. The pair might be red/green in old "quad" wire, or on the white/blue pair in new wire. It goes through the panel, through the normally closed contacts of a relay, when the panel is idle, and comes back out, via the jack, and thence back to the rest of the phone (or fax) equipment in the building on the yellow/black pair, or the white/orange pair. All the phones are "downstream" of the jack. That's to ensure that a fire in the building, even if it melts all the phones, will not hinder the alarm's ability to dial out and get help. It also will ensure that the cops will be called reliably if a burglar takes the closest phone to the door off-hook while he looks for the safe full of Krugerrands.
An experienced telephone tech will be able to state that a building does or does not have an alarm system, just be the presence or absence of this particular wiring scheme.
As my distinguished colleagues have stated, alarm panels can get stuck in their "off-hook" condition. That means that the relay inside the panel gets stuck. Being electronic, they are susceptible to power surges, low batteries, or evil spirits.
