In normal speech the word her, like other pronouns is normally not stressed. So in the following sentence, the word her would be unstressed:
Now the word her like other function words has a weak form. This usually has no /h/ and is just pronounced /ə/. (If it is the first word in the sentence the /h/ will be pronounced). So in the sentence above the word her is likely to be pronounced /ə/ in normal speech.
The word hear is normally pronounced /hɪə/ in non-rhotic accents such as Southern Standard British English. However, when the word occurs before a vowel, we will see /r/-liaison occurring. In other words we will pronounce the word with an /r/: /hɪər/.
The Original Poster's example is a bit more tricky.
John bought ten packs of star alliance stickers for her and him.
Here we have a co-ordination of third person singular pronouns her and him, where we would normally expect something like for them. Because the speaker is picking out each individual here, the most natural reading is that both pronouns will take contrastive stress. In other words it is very unlikely, if it's even possible, that her and him will be unstressed.
Because these words are stressed we need to use the strong forms of these pronouns and not the weak forms. In SSBE, this means that we need the forms /hɜ:r/ and /hɪm/ (the /r/ in her is present because of the following vowel). Notice that the strong form of her begins with the consonant /h/. Because of this, the /r/ in the word for, which occurs directly before her, will not be pronounced and will be realised just as /fə/.