It sounds like you have a fixture controlled by a conventional dimmer and are using CFL non-dimmable bulbs.
Standard dimmers were designed for incandescent bulbs. Many CFL and LED bulbs are non-dimmable. Some are dimmable, but only work with special dimmers designed for their electronic circuitry (conventional bulbs have no circuitry, only simple filaments). Some CFL or LED (not many) will work with a standard dimmer.
First you need to make sure the bulbs are dimmable, and then that you have a compatible dimmer.
In general, CFL and LED bulb packages indicate whether they are dimmable or not. If it does not say dimmable, it is probably not. Dimmers also indicate whether they are intended for CFL and/or LED bulbs. Again, if it doesn't mention CFL or LED, it is probably not compatible, unless you have bulbs that specifically say works with standard (or conventional) dimmers.
Check the fixture by using an incandescent bulb. If it works, either go with incandescent, find CFL or LED bulbs that work with standard dimmers or swap out the dimmer.
If the incandescent doesn't work, you have a wiring problem or a defective fixture.
Yes, dimmers reduce energy consumption of dimmable LEDs. Unlike incandescent bulbs, the electricity used is fairly linear with the light output; at 50% brightness it should use roughly 50% of the power.
Generally speaking, dimming will allow the bulbs to run cooler and extend their life spans. There may be some exceptions for some particular bulbs with particular kinds of dimmers, however, LEDs advertised as "dimmable" should not have any problems.
Best Answer
Given the symptoms, I have to suspect that this fixture is wired with the bulbs in series, rather than with the bulbs in parallel.
So, two incandescents - operate happily if somewhat dim and yellow on half-voltage. Bulbs would last a long time and veer towards "bug light" yellow without the ugly yellow paintjob, so it's not unreasonable for a "porch light" fixture.
One LED one incandescent - adequate voltage for the LED, so little current you don't even notice the glow on the incandescent.
Two LEDs - half voltage, ain't nobody happy.
Diagnostic - one bulb, one empty socket - what happens?