1) I heard him say that...

Please explain the grammar rules for "say" (present tense) after the word "heard".
2) A Economic Development Board survey...
Why is it "a" not "an"? Is it because it is "a survey"?
3) A historic moment...
I have seen some books writing "an historic moment"? It should be "a", isn't it?

1) I heard him say that...

Please explain the grammar rules for "say" (present tense) after the word "heard".
It looks like an infinitive form without the defining "to" -- in this case the infinitive would be modifying the direct object "him."

2) A Economic Development Board survey...
Why is it "a" not "an"? Is it because it is "a survey"?
This is an error. It should be "An..."

3) A historic moment...
I have seen some books writing "an historic moment"? It should be "a", isn't it?
Some people do; others don't:
http://owl.english.purdue.edu/owl/resource/540/01/
Scroll down and read "Remember, using a or an ..."

1) The grammar rule for using "say" (present tense) after the word "heard" is that verbs following "heard" are generally in the simple present tense. This is because "heard" is a verb that refers to an action or event that happened in the past but is reported in the present. For example, "I heard him say that he is going to the store." In this sentence, "heard" is in the past tense, while "say" remains in the present tense. This is because the act of saying took place in the past, but it is being reported in the present.

2) The reason it is "a" instead of "an" in the phrase "A Economic Development Board survey" is because the pronunciation of the word "economic" starts with a consonant sound (/ɪˌkɒnəˈmɪk/). When deciding whether to use "a" or "an," we consider the sound that follows the article. If the sound is a consonant sound, like in this case, we use "a." On the other hand, if the sound following the article is a vowel sound, we use "an." For example, "an apple" or "an hour."

3) The phrase "a historic moment" is more commonly used compared to "an historic moment." The reason for this is that the letter "h" in "historic" is generally pronounced with a strong "h" sound in modern English. Therefore, the word "historic" begins with a consonant sound, and "a" is used instead of "an." However, in some dialects or older versions of English, the "h" in "historic" may have been pronounced more like a vowel sound, resulting in the use of "an" before it. Nowadays, "a historic moment" is generally preferred over "an historic moment" in standard modern English.