If I had to guess, it would be that this form "{subject} {verb} {adjective} {infinitive phrase}" does not always work. The sentence you described, "I was happy to help you" will work but replace happy with other adjectives to see if it works.
I was hungry to help you
I was eager to help you
I was sad to help you
I was mad to help you
Out of all those sentences, the only one that actually made sense to me was "I was eager to help you", because it's just one of those common expressions. Similar to how "I was happy to help you" is a familiar phrase. But this doesn't mean that words such as sad and mad can not be used in the form "{subject} {verb} {adjective} {infinitive phrase}". For we can still say something such as
I was sad to hear that Craig Sager passed away today.
I was mad to see that question on the exam.
I was too tired to do my homework.
Like others have said, this most likely isn't a grammatical issue, rather, a semantics concept. It's related to why we wouldn't say "I am tired to eat right now" but why we would say "I am tired of eating cafeteria food".
EDIT: Seeing all the feedback on here has made me reconsider the validity of the sentence "I was tired to help you", so I hope I can congregate all the helpful feedback into my answer.
My four original examples
After reevaluating each sentence, I have to admit that each one was grammatically and logically correct.
I was hungry to help you
This can mean that I was desperate or extremely eager to help you.
I was sad to help you
This can mean that I was upset at the circumstances that lead you to needing my help.
I was mad to help you
This can mean that I really did not want to help you; therefore I was angry as a result for having to waste my time helping you.
Unfitting adjectives for certain indicative phrases
I agree with Mari-Lou A that adjectives such as possible are grammatically incorrect in the form "{subject} {verb} {adjective} {infinitive phrase}". The reason is that those kinds of words are usually structured with an infinitive phrase in this manner,
It is possible to grow onions there
It is unrealistic to assume that
It is imperative to hand your homework in on time
There are also adjectives that we can replace "tired" with that would not make sense either due to the lack of context or the lack of common meaning for the resultant phrase.
I was orange to help you
I was tall to help you
I was old to help you
What does it mean to be orange, tall, and old in these cases? Compare these sentences with
I was crazy to help you
I was foolish to help you
I believe that these sentences are all grammatically correct, but we only associate meanings with particular phrases. It may be possible that "I was tired to help you" and even the three sentences above all have legitimate meanings. However, I am not familiar with them because I have not heard these phrases used often.
The addition of too
An interesting point is that if you add too before tired, then the sentence seems to make more sense.
I was too tired to help you
Now in this case, being "too tired" has some common meaning associated to it. It means that I could not help you because I was tired at the time you required my assistance. We can add this word to our previous sentences to completely change their meaning
I was too hungry to help you
I was too upset to help you
I was too frustrated to help you
All these sentences have a common thread, which is the fact they all have the connotation that the action indicated in the infinitive phrase was never completed. So could it be that words such as tired can only express an incompleted action and that is why it requires the addition of too? I'm afraid this hypothesis has no validity, for some of Mari-Lou A's valid sentences included
I was anxious to meet her
I was impatient to meet her
Both these sentences do not guarantee that the action in the infinitive phrase actually occurred.
adjective + preposition + gerund
Another interesting point was how adjectives that work with the form "{adjective}{proposition}{gerund}" may not abide by the form "{subject} {verb} {adjective} {infinitive phrase}". The word tired falls into this category because of these two statements
I was tired to help you
I was tired of helping you
The second sentence means that I don't want to continue helping you. Other words such as afraid and bored fall into this category
I was afraid to help you
I was bored to help you
I was afraid of helping you
I was bored of helping you
It is a legitimate argument to claim that the first example, "I was afraid to help you" is completely valid. Unfortunately the second example, "I was bored to help you" falls into the category of less familiar sentences such as "I was tired to help you". So the gerund topic did not get us very far.
My final thoughts
What does it mean "to be tired to help someone"? Can we replace tired with a synonym to get a similar sounding sentence that is more commonplace? Exhausted, weary, and fatigued don't seem to work either, or do they? Looking at the sentence, "I was happy to help you", I tried to decipher what it meant. It simply means that I wanted to help you because it gave me satisfaction. Could it be possible that "I was tired to help you" has a meaning that we are not fully aware of? I only asked this because users were arguing about the meaning of phrases with questionable adjectives like "I was annoyed to see her" and "I was mad to see her". It may very well have to do with why "I was not there to help you" works but "I was absent to help you" doesn't. All of the examples could be grammatically correct but we don't know what they all indicate.