Short answer: Maybe.
Long answer: Depends on how "fringe" your fringe area is.
CDMA generally doesn't degrade gradually as the situation gets worse. Rather, it continues to operate nominally up to a point, then degrades quite rapidly. It's sort of a cliff. This is quite unlike for instance AMPS (the old analog cellular system). As signal strength degrades, AMPS audio gets gradually worse in a fairly linear fashion. As signal strength degrades with CDMA, there is (or should be) no noticeable change in the sound quality until it suddenly gets dramatically worse and the call drops.
So it is with CDMA idle mode. Begin with the fact that cell sectors are often somewhat small and that you can carry your phone from one to another quite easily. When this happens, the phone has to switch from one to another so that it can keep listening for an incoming page, which indicate that you have a phone call coming in.
Since the phone doesn't have any way to keep track physically of where it is, and doesn't know the physical layout of the cell system in any case, what it does is monitor the signal strength of the cell it's on. As long as that strength is adequate to permit highly reliable reception of pages, the phone is happy and sits in what is known as slotted sleep, a mode of operation which is extremely efficient in use of energy.
But suppose that you get in your car and drive from one cell to another. As your phone moves away from the first cell, the signal strength will begin to fade. Eventually it will reach a point where reception of pages is no longer reliable. Sometime before that point, the phone will begin to search to see if it might be able to find a different cell or sector which might have a stronger signal. If it didn't do this, once it left the first sector you'd be out of contact with the cell system and would no longer receive incoming calls.
This process is called searching, and in a well built-out system it ordinarily doesn't take long for the phone to find a new cell. (That's because the cell tells the phones periodically which neighbors to look for on the paging channel. So the phone has a limited list of PN offsets to check to look for neighbors.) Once it finds a better one, it performs an idle handoff which means that it begins to listen to the new cell instead. (If the new cell is in a different zone than the old one was, the phone will turn its transmitter on briefly and register with the new cell, announcing to the cell system that it has moved. If it remains in the same zone, registration is not needed.)
The point I'm trying to demonstrate is that searching is necessary in order to make your mobile CDMA phone truly mobile. The problem is when you happen to be sitting still in an area where signal strength is poor, such as at the very edge of the system.
The phone can't easily differentiate this case from the previous one. What it knows is that the signal strength of the cell it's been listening to isn't sufficiently high. So it listens to try to find another cell, but it can't find any alternatives either. Oh, well, try again later – a few seconds later.
All that searching requires keeping the receiver in the phone on, which means that power is being consumed at a much higher rate than in the normal idle mode. The phone has to keep doing this, because it has to try to find a better cell if it can.
Now this behavior definitely depends on a threshold, and thus you can get big differences in results from small changes in the environment. The same phone in the same location can act differently at different times of the day. Two phones at the same location may act differently depending on how sensitive their receivers are. Moving the phone as little as 6 feet can change it, because microwaves are highly linear and you might move into or out of a shadow of some object between the phone and the cell tower. (Some people have reported that they can lose the carrier when in the drive-through line at a hamburger stand. That's because of the metal stoves or large refrigerators inside the store, which act as excellent shielding. If placement of the phone, stove and antenna are just right, the signal level can drop precipitously.)
So whether or not your battery gets chewed up like this depends critically on just how bad the signal is at the location where you operate your phone.