I did that before, when the circuit was on a breadboard. But today, seeing as the circuit is soldered and I didn't have a soldering iron available, I couldn't. I used to the method described on this website.
http://www.kpsec.freeuk.com/components/tran.htm

 

Testing a transistor with a multimeter can show that its probably Ok.
But, testing the transistor in your circuit ensures that everything is working as intended, including the wiring.

I did that before, when the circuit was on a breadboard.

If you tested the 2n3904 as suggested in-crcuit and everything worked as expected, then the issue has to be with the detector.
According to the datasheet, the detector has an internal pullup resistor on the output and it sinks the output low when IR is detected.

So-- No IR, the output should be at V+, IR present, the output should be near gnd. The post earlier stating that the detector output never goes above 0.488v is a Red Flag - detector not working or mis-wired.

 

When the detector isn't hooked up to anything else and I measure the output with no IR light, it was at 5V as it is supposed to be. The output went low (about .04V) when IR was detected. Then, I added the 10k resistor, 2N3904, and pull-up resistor and measured the output voltage again. The output was the same (.04V) when it was sensing IR, but without IR the output was about 2.1V.

I have 4 of the IR detectors. Each one was the same.

 

That's better than the previous post. If you are measuring the 'output' at the detector with the 10k base resistor then its about right when you consider the detectors internal pullup in series with it
Now check the collector of the 2n3904. It should be opposite -- near gnd with no IR and at V+ with IR detected. And, you do have a 10K resistor in series with the collector to V+ correct?

 

I do have a 10k resistor in series with the collector to V+. I don't have a way to check it right now, but when I tested it earlier the collector of the 2N3904 was low when there wasn't any IR. When there was IR, I'm fairly certain the collector voltage didn't change, and that's where I was confused. I was thinking its possible the 2N3904 was destroyed somehow during earlier testing, but checking that today with the circuit on the site I posted earlier seems to show that it is fine.

Also, I remember when the circuit was powered and the detector wasn't connected, the 555s ran as expected. This created the situation of the detector output being low, so those worked properly.

 

I don't have a way to check it right now, but when I tested it earlier the collector of the 2N3904 was low when there wasn't any IR. When there was IR, I'm fairly certain the collector voltage didn't change, and that's where I was confused. I was thinking its possible the 2N3904 was destroyed somehow during earlier testing, but checking that today with the circuit on the site I posted earlier seems to show that it is fine.

555's resets need to be pulled below 0.7 vdc to reliably reset or stop the timer. If they are at that voltage or higher they usually run. So if the transistor is marginal that may be the problem. I'm understand that the timers run but never shut off - correct?

You could just jumper the 2n3904/RST pin connection to ground to see if the timers stop.

Also, I just noticed that pin 5 on the bottom right 555 is un-connected. Put a 0.1uf cap from that pin to gnd. Leaving that pin unconnected can cause troubles sometimes.

 

When the detector is connected to the circuit, the timers are off, even when IR light is present. However, when I disconnect the detector, the timers run. They should run when the detector is not connected as it simulates a low output voltage like the detector is sensing light.

The timers are not constantly running.

 

When the detector is connected to the circuit, the timers are off, even when IR light is present. However, when I disconnect the detector, the timers run. They should run when the detector is not connected as it simulates a low output voltage like the detector is sensing light.

The timers are not constantly running.

This symptom shows that either the detector(s) aren't working or your IR source isn't at the correct frequency or even working.

Try removing the base resistor connection from the detector and connect it first to V+, then to gnd to simulate what the detector is supposed to do. That will prove correct operation through the transistor, leaving only the detector as the culprit.

Then test with the detector connected using an IR remote.

The detectors may have been damaged along the way, but it may simply be an incorrect IR signal.

 

I checked the IR LED with a cell phone camera to make sure it is on, and it was. I verified the oscillation frequency and it was correct, right around 36.7kHz.

An IR remote has no effect on the operation of the circuit.

I don't have a soldering iron handy, so I can't say what happens when the base resistor is connected to V+ and then GND.

Another thing I found weird is that when the battery is first connected, an LED set up to blink using a 555 flashes briefly.

 

I don't know about your Japanese IR receiver IC because the translation in its datasheet seems to be missing all the details.
The Vishay IR receiver IC turns down its gain (very low) when it receives continuously modulated (36kHz) IR because it is probably interference from a compact fluorescent light bulb. The Vishay IC needs bursts of 36kHz IR with certain numbers of pulses followed by a certain pause then more pulses, etc which is the data transmitted by a remote control.

 

I see your point Audioguru, but when I set up another detector and exposed it to the IR light coming from the soldered circuit, the detector acted as expected.

 

An IR receiver IC that reduces its gain when it receives continuous IR interference from a compact fluorescent light bulb still receives some pulsed IR if it is bright enough or is close enough.

Why don't you use an IR receiver IC that has a detailed datasheet written in good English instead of using that Japanese one?

 

This is a project for a class, and we were told to use the PNA4611M.

 

It is too bad that Panasonic's English datasheet is missing all the details that are on all other English datasheets.
Maybe your teacher knows nothing about IR receiver ICs.