If a person has an illness, lives in fear or pain and cannot work, or doesn't have the strength to work, does 2 Thessalonians 3:6-12 condemn that individual?
This passage does not specifically mention the ability to work, but the desire to work.
The portion in question you are mentioning is specifically the phrase "that if any would not work, neither should he eat." Here is the Greek for that phrase:
ὅτι εἴ τις οὐ θέλει ἐργάζεσθαι μηδὲ ἐσθιέτω
I have emphasized the word which we translate "would". In English, we have the word "will", which is ambiguous. It can either mean to desire or will to do something, or it could be used as a helper verb to indicate tense. However, in Greek, there is no such helper verb. It simply means to desire or will to do something.
The difference is that, unlike the English, it does not imply a lack of future action, but instead a present lack of willingness. A person who is unable to work may or may not be willing, regardless of his ability. If he is willing, then this verse does not apply to him.
A further clue can be found in verse 11:
For we hear that there are some which walk among you disorderly, working not at all, but are busybodies. (KJV)
Here is the Greek:
ἀκούομεν γάρ τινας περιπατοῦντας ἐν ὑμῖν ἀτάκτως μηδὲν ἐργαζομένους ἀλλὰ περιεργαζομένου.
It is an interesting play on words, like Paul seems to do so often. He's basically saying, "There's some walking around, who don't work, but around-work instead". The word "busybodies" is "περιεργαζομένους", which is literally, to "work about/around", meaning to go about meddling in other people's business. You can see here that the warning is not to be sure that people are working, but that there are people who are spend their time and energy on pointless or hurtful things, rather than on being productive.
If you can work, work. If you can't work, then you also can't go about meddling. Whatever your abilities may be, devote those unto the Lord.