Ge 15312 will not turn on by remote switch

1) Guessing you have 3-way switch where 2 different switches control same Load (light, fan, motor). Unknown wiring, and unknown if fan and light are wired separately.

2) Google search for 3-way fan switch shows Leviton IPF05 with simple straight-forward wiring. Link below has instruction manual, and Live chat button located to right of LEVITON
http://www.leviton.com/OA_HTML/ProductDetail.jsp?partnumber=IPF05-1LX&section=47082&minisite=10251

3) It appears that the Leviton device must replace Line side switch.
http://waterheatertimer.org/images/GE-15312-3-way-circuit-600.jpg

4) Uncertain if this device also controls fan since the wiring is so simple.

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There are 2 different timers: 15313 and 15312
15313 has Black Blue and Green wires
15312 has Black Red White and Green wires

1) Your Switch has 3 wires: sounds like a 3-way switch at each end of hallway.
The 15313 timer cannot be used to replace 3-way switch
http://waterheatertimer.org/GE-timers-and-manuals.html#15312

2) Buy GE 15312 timer for 3-way switching.
http://waterheatertimer.org/How-to-wire-GE-15312-timer.html#3-way

3) GE 15313 is for single-pole only, and rated for incandescent lights only and will not work for CFL, LED bulbs, or other inductive loads.

4) GE 15313 timer will work for single-pole or 3-way, and work with any type of bulb, or load up to 100 feet away.

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Copy following link for replacing single-pole switch with single pole 15086 / 15071 GE timers.
http://waterheatertimer.org/How-to-wire-timers.html

Copy following link to identify GE timer:
http://waterheatertimer.org/GE-timers-and-manuals.html

Copy following link for GE 15312 single-pole and 3-way timer
http://waterheatertimer.org/How-to-wire-GE-15312-timer.html

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

It sounds like you are replacing 3-way switch.
Low voltage light? Are these lights connected to 120Volt circuit?

I think you want the GE 15312 timer, instead of GE 15313.
http://waterheatertimer.org/How-to-wire-GE-15312-timer.html

GE 15313 is for single-pole incandescent light bulbs only.

This is live answer 11-2012
Add a comment for more interaction on answer.

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10-26-12
Always clear digital timer programming using reset button.

Wiring:
Copy following link to double-check wiring is correct:
http://waterheatertimer.org/How-to-wire-GE-15312-timer.html#3-way

Scroll down and look at each of the three 3-way illustration, Line side, load side, without neutral.

Here's how the timer works for 3-way:
When the red wire receives power, it reverses the timer programming of on-becomes-off or off-becomes-on.
So the purpose of the remaining 3-way switch is to connect-or-disconnect power to timer red wire >> this causes the 3-way effect.
Basically the remaining 3-way switch is converted into a simple on-off switch by using only one of the brass screws, and the dark screw.
So one of the brass screws on the remaining 3-way switch has no wire connected to it. It makes no difference which of the two brass screws is selected.

Add a comment as needed.
This timer has 1 year limited warranty through jasco, or just buy another one from Home Depot and return defect in new package.

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Check circuit breaker.
Test if line has power.
Check light bulb.
Neutral wires loose under wire nut.
Switch wired incorrectly?
Copy and paste following link to see basic single-pole and 3-way switch wiring:
http://waterheatertimer.org/How-to-wire-GE-15312-timer.html

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

Copy following link for timer wiring.
Your box does not have neutral wire
The GE 15070 7-day timer wires same as GE 15312 7-day sunsmart timer
http://waterheatertimer.org/How-to-wire-GE-15312-timer.html

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

Yes 100' maximum is problem. To work around problem, you can use timer to turn on a contactor, and then the contactor is not limited to 100' distance.
Copy following link to see wiring diagram, and you can wire it yourself:
http://waterheatertimer.org/images/Dawn-dusk-120Volt-pump2-300.jpg
http://waterheatertimer.org/H230A.html

Assuming timer was installed correctly ... did timer work when first installed, and then die?
This timer cannot be exposed to freezing temperatures because the battery will die in cold temperatures.
Ei600 timer is 5-year warranty version of ST01C timer, and both have same wiring and same battery.
Copy following link to see programming, wiring, and battery replacement
http://waterheatertimer.org/ST01C-program.html

If cold battery is problem, suggest using GE 15312 timer from Home Depot that has same 100' maximum and same load rating characteristics and same programming capability as Ei600 and ST01C, but GE timer is powered by the circuit and not a battery.
http://waterheatertimer.org/How-to-wire-GE-15312-timer.html
You can install timer yourself.

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

In residential wiring, it is VERY unusual to have 2 hot wires connected to one switch.
The exception is when switch controls 240Volt load, and turns off both hot wires:
Copy following links to see example DPDT switch with two hot wires.
http://waterheatertimer.org/images/30-Amp-DPDT-switch-a-500.jpg
http://waterheatertimer.org/How-to-wire-water-heater-for-120-Volt.html
Copy following link to see typical switch wiring diagrams:
http://waterheatertimer.org/How-to-wire-GE-15312-timer.html

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

Woods 58018 is White-colored 120Volt 7-day programmable timer that replaces single-pole switch only.
I am not certain what is meant by 3-pole circuit.
If you mean 3-way circuit where more than one switch control same light, as you find with hallway light, then 59018 is not correct timer.

Open following links to see list of timers suited for 3-way switch replacement.
http://waterheatertimer.org/Woods-timers-and-manuals.html#59028
http://waterheatertimer.org/GE-timers-and-manuals.html#15312
http://waterheatertimer.org/ST01C-program.html#ST01C

Timers that work for 3-way do not replace 3-way switch wire-for-wire.
Read manual carefully.
When you replace 3-way switch with a timer, one 3-way switch is replaced, and other 3-way switch remains.
The remaining 3-way switch is always rewired so the 3-way timer works. AND each brand of timer requires slightly different wiring parameter.
Some 3-way timers can only replace the 3-way switch located closest to circuit breaker box, commonly called the LINE side. Others can replace either LINE side or LOAD side 3-way., and wiring is different for each. Some timers require neutral wire and some are battery-powered.

Add a comment with exact information about switch and wiring you are working on for suggestion of best timer for that purpose.