Different languages have different rules about how to use commas. You cannot use the punctuation rules of your native language when you write in English – you must use English rules.
Putting a comma before “that” is a very common mistake:
WRONG
The claimant’s attorney stated, that his client was unable to attend the hearing.
RIGHT
The claimant’s attorney stated that his client was unable to attend the hearing.
In this example “that his client was unable to attend the hearing” is a subordinate clause. Languages such as German, Czech and Polish have the rule that a comma should be inserted before every subordinate clause. In English we do not have this rule (if we did, I would have put a comma between “rule” and “that” in the previous sentence).
This issue is not limited to subordinate clauses beginning with “that”. Here are some other examples:
WRONG
Please review the draft to see, if it fully reflects your intentions regarding employee remuneration.
A VAT obligation arises, when the goods have been delivered.
The company should check, whether the contractor has the required experience.
RIGHT
Please review the draft to see if it fully reflects your intentions regarding employee remuneration.
A VAT obligation arises when the goods have been delivered.
The company should check whether the contractor has the required experience.
This sort of mistake matters in English because it can be very confusing for the reader. In English commas are used to mark places where the reader naturally pauses, or they are used to “section off” parts of sentences.
There is no natural pause before the subordinate clauses shown above, and there is no reason to section off the subordinate clauses from the rest of the sentences. Therefore such commas are incorrect.
Also – native English speakers never put commas in such places, so by doing so you’ll be giving a clear indication that English is not your first language.
Common Mistakes in Business English 
How about examples where comma before that is legitimate, e.g. when it’s one of the commas “sectioning off” a comment?
I don’t think a comma before “that” is ever legitimate. However, a comma before “which” can be. Check out my post “How to use commas part 4” – the section titled Bracketing commas with relative clauses.
Actually, wouldn’t a comma be correct if it’s not a conjunction?
Example: “They tend to drink a lot of booze, that means you often cannot talk to them.” since “that” just refers to the whole first part?
Thanks for your message Alex. Interesting comment!
Your example is incorrect because “that means you often cannot talk to them” is a non-defining relative clause. It is not usual to start such clauses with “that”; instead we use “which”. You could use “that” only if it starts a new sentence – “They tend to drink a lot of booze. That means…” So the comma before “that” is still incorrect.
Thanks a lot for the clarification!
Namename – Actually you may be right, if your comment refers to examples such as the one Mustafa gives below. A comma before “that” can be legitimate when it is a bracketing comma “sectioning off” some words.
It’s archaic or poetic I guess, the use of ‘that’ that way. Here’s an example from Nine Stories by J.D. Salinger: It was the move back to New York, nine years later, three months after my mother died, that threw me, and threw me terribly.
Thanks Mustafa, that’s an interesting quote. But actually the comma is fine there. It is a bracketing comma which (along with the comma before “nine”) sets apart the words “nine years later, three months after my mother died”. Bracketing commas show that the sentence will still make sense if the words set apart by the commas are removed – like this: “It was the move back to New York that threw me, and threw me terribly.” It would be incorrect to put a comma before “that” in this case. For more information on bracketing commas see my post on the subject: https://blog.harwardcommunications.com/2015/11/17/how-to-use-commas-part-4/
I see your point, Mr. Harward. You are probably right.
How about this: This was such a new idea to Alice, that she was quite silent for a minute or two. (Alice’s Adventures in Wonderland by Lewis Carroll)
That’s an interesting one, Mustafa. But I think nowadays writers would not include a comma there. The pattern “it was such a… that…” (and variations) can be used in lots of sentences, and it’s not usual to add a comma. For example:
It was such a hot day that she stayed in the house and closed the shutters.
He sat and waited for such a long time that when he stood up he had cramp in his leg.
It was such engaging work that she had no idea so much time had passed.