Greetings Olivia & others..
First of all, what kind of test equipment do you have use of? Do you know how to use it? If not, you're going to need to find someone that IS familiar with anything from a DVM (Digital Volt Meter) to, ultimately, an oscilloscope and has use of them.
Regarding: "There appeared to be only one burnt out component and I replaced it but it still doesn't work." What specific component was that? Further, does it relate to the schematic of the .pdf file referenced?
"I'm pretty certain that this is a copy of the Boss TR-2 because the schematic seems to fit with the exception of the IC chips."
"It did look like something was spilled inside the pedal and I've cleaned that out." Any idea what was spilled in it? Was it sticky? If it was something like a soft drink that has sat for some time without being cleaned out, there's a good chance that the potentiometers may also need to be replaced in spite of a electronic component failure. I've had such experience fixing a fellow radio amateur's HF radio where his daughter spilled Coke all over the front of the radio. It was a mess and a good deal of labor to resurrect the front panel alone, not to mention the rest of the radio!
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It would probably be a good idea to actually get the schematic to the piece of electronics that you've got. Writing Johnson may provide this, request a service manual -
Keeping in mind that this REALLY isn't the correct schematic for the unit you have, in order to resolve this, you're going to first need to verify that you've got B+ to the electronics. The power supply circuit is at the bottom of the schematic above AC ADAPTOR. Namely +9V, +5.5V & +4.5V DC should be verified to exist. NOTE: Depending on what was burnt out will also dictate where one should also be looking. If any of these biases are missing, they will have to be fixed before you can do any further diagnosis!
Next, with the above voltages present, turn OFF any and all special effects and apply a signal into the input and follow it through with either a DVM or, ultimately, an oscilloscope towards the output. Incidentally, that signal path initially passes through R15, C8 to the source of FET transistor Q4. Continue to follow that signal towards the output through either the IC1B, IC2A, Q1 or the Q3 path all leading to the positive lead of C1 (1uF/50V). Finally through Q2's base to the emitter and through C2 & R1 to the output.
If you lose the input signal anywhere along this path, back up to the component or region where that signal will not pass and verify that the signal exists on one side but not the other. Attempt to isolate to your suspect component. Verify correct DC bias to the component, if it's still not passing a signal, that will be the component to replace.
As you get the direct (non-effect) path for the pedal to work, then you can look into the individual circuits for the Rate, Wave and Depth effects according to the picture at the above URL. By using the same method of tracing a signal from the input of the effect electronics through towards the output of that circuit, one by one, you'll verify each subsection of the circuit.
Trying to learn all of what you need in a Use-group reply or two is usually not adequate. Again, having someone that is versed in electronics theory, diagnosis and repair that either has use of or owns their own test equipment will substantially reduce the amount of effort and time you'll end up dedicating for, essentially, a $60/USD piece of electronics.
Good luck!
Cheers, Mr. Mentor