Please be aware that the pronunciation could vary from one dialect to another.
However, the words you mentioned seem to be pronounced pretty much the same in both American (AmE) and English (BrE) accents. One main difference here is that BrE is typically non-rhotic ([r] sound is more gentle and subtle), while AmE is typically rhotic.
heir is pronounced "air"
hare and hair are pronounced "hair" (with the same vowel as "air")
here rhymes with "ear", but with "h" sound
hire rhymes with "fire", but with "h" sound
her is pronounced "her"
Your dictionary should provide another good way to check the pronunciation.
Finally, thanks to @Hellion, in listening, to tell which word is which from several possible homophones (e.g. heir or air), context will give you necessary clues.
Answer from Damkerng T. on Stack Exchangepronunciation - How to distinguish words hare, here, hair, heir, hire, her, higher, hear? - English Language Learners Stack Exchange
Hair-Hare [pronunciation] | WordReference Forums
Do any English dialects contrast the words "hair" and "hare"?
What is the difference in pronunciation between
heir and air
hair and hare?
Videos
Please be aware that the pronunciation could vary from one dialect to another.
However, the words you mentioned seem to be pronounced pretty much the same in both American (AmE) and English (BrE) accents. One main difference here is that BrE is typically non-rhotic ([r] sound is more gentle and subtle), while AmE is typically rhotic.
heir is pronounced "air"
hare and hair are pronounced "hair" (with the same vowel as "air")
here rhymes with "ear", but with "h" sound
hire rhymes with "fire", but with "h" sound
her is pronounced "her"
Your dictionary should provide another good way to check the pronunciation.
Finally, thanks to @Hellion, in listening, to tell which word is which from several possible homophones (e.g. heir or air), context will give you necessary clues.
How can I distinguish between the words 'hare', 'here', 'hair', 'heir', 'hire', 'her', 'higher' and 'hear' when pronounced?
Context. Here's some hints that can help you.
hare isn't too common of a word unless the conversation involves rabbits or animals, the expression hare-brained being a possible exception. So you can assume "hair" if you hear this most of the time.
hear is a verb and will be immediately preceded by a subject pronoun most of the time - I hear X... etc. here will not be immediately preceded by a subject pronoun. hear can be the first word of a sentence, but not too often - using hear in the imperative sense, where it would be the first word of the sentence, would be limited to an announcement scenario, or possibly where someone is telling you to listen closely.
heir isn't a common word outside of a royal or legal setting. Nine times out of ten this sound is the word air.
hire is a verb and will be immediately preceded by a subject pronoun most of the time, or be used in the expression for hire. Higher, being an modifier, will not. EDIT: hire is also a noun, typically used in the phrase new hire or recent hire.
her should not sound like the other words, but this could be the case in some dialects. Usually her will occur towards the end of a sentence and preceded by a preposition like "to" or "by." If it occurs towards the beginning of a sentence, it will be immediately followed by a noun, expressing possession - "Her face was beautiful."
There are a few English dialects in Wales (Abercraf and Port Talbot) that contrast the words pain and pane (realising them as [æj~ej] and [eː]). These dialects avoided the pain-pane merger that occurred late in Middle English. However, I can't find out if this contrast is also made before /r/, because Wikipedia says that these dialects have only one realisation of the SQUARE vowel, and Google isn't helping very much, so I don't know where to find this information.
Thanks in advance!