Not necessarily. If the station has a production studio, then that can be moved first; a spare office computer and maybe even a laptop can be rigged to run any digital storage system for a day or so.
The station's engineer (likely a contract engineer) can get an electrician to pre-wire any special studio needs and to get any required permits. A small station is not gong to need a lot of wiring other than Cat5/6 cables between studios and the interconnect point in stations and the TELCO and ISP access points. If the equipment is old, then conventional audio cables can be put in too.
If the station is not using more modern digital audio handling inside the building, this is the time to convert as the other gear will become increasingly obsolete.
The cost may be just laying cables and having them ready for carrying the equipment, which could be done overnight even if there is no production studio to use temporarily. I'd say the most complex portion is moving the digital storage / automation if there is no backup computer that can be moved without disturbing the on-air one.
I've seen fully automated stations carry a duplicate computer to the transmitter site where it runs the whole station over a weekend while the physical plant is moved.
Good point. It may be that the connection is via a Barix box or similar, which is easier. In any event, using leased audio lines is something that should be abandoned in favor of a different technology. There are fewer and fewer phone company techs who know about lines, so eventually that is a dead end.
Another consideration for the future is whether the building has, or can easily be wired for newer technology. Looking for a smart building is important, and that included finding out if the neighborhood has fiber pre-wiring, capacity, etc.
The station contract engineer should sit down with the owner and make a checklist of all these issues.