I cannot program the "Off" time on my GE smart digital timer model 15071

Just replace it. They wear out, even the electronic type.

https://www.ebay.com/p/1600036286

https://www.google.com/search?client=firefox-b-d&q=Ge+15071+smart+time

They also have an internal battery though, which does wear out. I found this on Fixya. this is most likely your issue.

The battery is build into circuit board and not replaceable.
The following 2 links have resources for 15071 programmable timer:

http://waterheatertimer.org/How-to-troubleshoot-timers.html#in-wall

http://waterheatertimer.org/Programmable.html#15071

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GE 15071 sold at Home Depot and on-line has feature that adjusts for daylight savings time.
GE 15086 does not offer automatic adjustment for DST
http://waterheatertimer.org/GE-timers-and-manuals.html#15071

Daylight savings time (DST) is not same as dawn-to-dusk astronomical timer.
GE 15071 and 15086 single-pole timers do not have dawn to dusk astronomical feature.

For single-pole dusk-dawn astronomical with DST, Lowes sells Utilitec-0192773
http://waterheatertimer.org/Program-wire-Utilitec-0192773.html

For 3-way switching, 7-day DST -or- dawn-dusk astronomical:
http://waterheatertimer.org/Aube-timers.html

Open following link for GE 15071 instructions, image and manual
http://waterheatertimer.org/GE-timers-and-manuals.html#15071

GE 15071 is single pole only.

Open following link to see GE 15071 timer, on-line manual, and steps for wiring:
Link also shows timers that work for 3-way switch and include links to those timers and manuals:
http://waterheatertimer.org/GE-timers-and-manuals.html#15071

Add a comment for more free help.
Also take advantage of fixya phone service.
For a price, expert speaks with you over phone while you work on timer or any do-it-yourself project.
Fixya is always less expensive than a service call.

Timer wiring instructions shown on following page:
http://waterheatertimer.org/GE-timers-and-manuals.html#15071

You connected the white wire and red wire together, and that is why lights are clicking on-off.
I test timers and have done this intentionally during testing.
Follow instructions shown on link above, and GE timer will work correctly.
Same wiring applies to Tork, Swilight, Utilitec, Woods, Leviton, Intermatic non-battery operated timers.
Instructions are different for Aube, Brinks, UPM and Intermatic battery operated

I am guessing that you have digital in-wall timer GE 15071 or GE 15086

To identify timer for sure, see images at following link:
http://waterheatertimer.org/GE-timers-and-manuals.html#15071

Manual for 15086
http://waterheatertimer.org/pdf/GE-15086-Manual.pdf

If this is new timer, the problem could be a couple things.
1) Circuit has electrical problem that is affecting timer electronics.
Use circuit analyzer from Home Depot electric department and test any receptacle on same circuit breaker as timer.
2) event1 ON was programmed but event1 OFF was not programmed. To review events, press Prog button once to see event1 ON and press Prog button again to see event1 OFF. When review is complete, press clock key to return to current time.
3) event1 ON and event1 OFF are programmed with conflicting day schedules
4) event1 and event7 have conflicting schedules

If this is new or old timer:
5) Timer needs to be reset because of internal glitch. Press and hold reset button for a moment and timer display will start flashing 12:00 am MO
6) These timers can go bad any time. I test timers and 4-6% arrive defective, and most quit after couple years, but some can last quite a while. This is why I suggest using a circuit analyzer to check if circuit is causing difficulties.

You posted under GE timer.
Open following link to identify timer and see manual and written wiring instructions:
Most in-wall timers are wired the same way except timers listed on page that work for 3-way switching.
http://waterheatertimer.org/GE-timers-and-manuals.html#15071

Add a comment for more free help.
Also take advantage of fixya phone service.
For a price, expert speaks with you over phone while you work on timer or any do-it-yourself project.
Fixya is always less expensive than a service call.

Your timer is bad. It comes with 1 year limited warranty.
Buy one at Home Depot and then return defective one.
Make note of wiring before removing bad timer, so re-wire is easy.

GE 15086 is White 7-day digital, non-astronomic timer made by Tork, and sold on-line and at Home Depot.
On the shelf, 15086 timer looks identical to 15071 >> same package >>> difference is 15071 offers daylight savings time and random option, and costs a bit more.

These in-wall timers usually last a few years, and some can last for several years.
Especially when timer operates from electric circuit and not battery.
15086 operates from electric circuit.

15086 manual on-line:
http://waterheatertimer.org/pdf/GE-15086-Manual.pdf

The timer is wired incorrectly.

ON the GE timer and Utilitec timer: The red and black wires are reversed.

On the Intermatic ST01C

Sounds like you connected white wire incorrectly, or are trying to replace 3-way switch.
GE 15071 and 15086 are for single-pole switch replacement only, and will not work for 3-way.
Open following page for correct wiring of GE in-wall timers, and for list of timers used for 3-way.
http://waterheatertimer.org/GE-timers-and-manuals.html#15071

If you have different timer, add a comment with in-focus photo of your timer for correct identification and possible assistance.
http://www.fixya.com/support/r6568559-post_photograph_fixya_questions_answers

Add a comment for more free help

Also take advantage of fixya expert assistance live.
For a price, expert works with you while you work on timer or any do-it-yourself project.
Fixya is always less expensive than a service call.