First of all, Google can find quite a lot of background on the dispute between Steve Lipsky and Range Resources Corp. This article is a summary of the situation, where Range Resources Corp. has been drilling for gas close to Lipsky's estate, after which Lipsky is blaming the company for ruining a well, from which he used to get his fresh water supply. It is perhaps also worth to note that Lipsky has been convicted for being 'part of a "conspiracy to defame" the company because he'd circulated a video "calculated to alarm the public into believing the water was burning."'. Assuming that the verdict was correct, Lipsky has previously distributed videos with fraudulent claims about the water from his well burning.
The news article I linked to, mentions methane contamination as the reason for the burning water, which is consistent with the fact that Range has been drilling for natural gas nearby. Methane is indeed a highly inflammable gas (mixed with air, the methane content must exceed appr 5% for the mix to be inflammable) and it seriously contributes to the greenhouse effect, but except for that, it is pretty harmless with no known toxic effects (material data safety sheet). There are even relatively high levels of methane in the humane digestion tract. According to this study, the average flatus (fart) has a methane content of about 7%.
Now, methane is barely soluble in water, according to Wikipedia, only 22.7mg/l. If you look closely at the video from 0:20 to 0:30, there is also no obvious proof that the water is actually burning. To me, it looks as if a gas escaping the water pipe is burning at the opening. If the water had actually been burning, why wouldn't the water jet burn further away from the pipe opening as well?
So to answer your questions:
Is it safe to drink water, which can be set on fire?
In this case, there is no sign that the water is actually burning. If whatever burns at the pipe opening is methane gas, it is in no way harmful to drink water having been in contact with methane gas or with small amounts of methane disolved.
Is it for ordinary home purposes safe to use water, which can be set on fire?
Still doubting that the water is actually burning, what may be a problem is that methane is released in closed rooms through a water pipe network. Even here, small levels are uncritical, but depending on how fast methane is released and how fast the air in the room is replaced, there may of course be a risk, that explosive levels of methane are reached.