'if' and 'whether' are more or less interchangeable in sentences like, " I'll see if he left an address" and "I'll see whether he left an address" although 'whether' is generally regarded as more formal and suitable for written use. But, although 'if' and 'whether' are often interchangeable, one distinction is that 'if' is also used in conditional constructions and 'whether' in expressing an alternative or possibility. Thus, "Tell me if you’re going to be in town next week" could be strictly interpreted as 'you need not reply if you are not going to be in town,' whereas, "Tell me whether you’re going to be in town next week" clearly means 'a reply is desired one way or the other.'
'whether' is used when we are expressing a very real doubt or choice between alternatives As to this use with 'doubt', the following is from American Heritage Word Uage:
'doubt' and 'doubtful' with 'that', whether', or if. You can follow doubt' and 'doubtful with clauses introduced by the conjunction 'that, 'whether', or 'if'. Which conjunction you choose depends somewhat on the meaning you want to convey. We normally use 'whether' to introduce indirect questions: I asked whether he could come along. 'Whether' is therefore the traditional choice when the subject of doubt is in a state of genuine uncertainty about alternative possibilities: Sue has studied so much philosophy this year that she’s begun to doubt 'whether' she exists. Similarly, when 'doubtful' indicates uncertainty, 'whether' is probably the word you want: At one time it was 'doubtful' 'whether' the company could recover from its financial difficulties, but the government loan seems to have helped. 1
On the other hand, you use 'that' when you use 'doubt' as an understated way of expressing disbelief. Thus you might say : I doubt that we’ve seen the last of that problem, meaning “I think we haven’t seen the last of that problem.” 'That' is also the usual choice when the truth of the clause following doubt is assumed, as in negative sentences and questions. Thus : I never doubted for a minute 'that' I would be rescued implies “I was certain that I would be rescued.” By the same token : Do you doubt that you will be paid? seems to pose a rhetorical question meaning “Surely you believe that you will be paid,” whereas : Do you doubt whether you will be paid? may express a genuine request for information and might be followed by 'Because if you do, you should make the client post a bond'.
In other cases, however, this distinction between 'whether' and 'that' is not always observed, and 'that' is frequently used as a substitute for 'whether'. You can also use 'if' as a substitute for 'whether', but 'if' is more informal in tone.