It is a situation which a salesperson announces to a group of people to attract them to buy his goods.
Here is the full sentence.
You can add focus particles like は or も to verbs, but in order to do so, you have to split the verb into two parts so that the particle has some place to go. We'll split the verb into its continuative stem (called 連用形 in Japanese) and the verb する.
忘れる → 忘れ＋する 忘れる＋も ＝ 忘れもする
忘れない → 忘れ＋しない 忘れない＋は ＝ 忘れはしない
Your example is a little more complicated for a couple reasons. First of all, ある has a suppletive negative form. Instead of saying あらない, people just say ない, almost like the
あら has been deleted:
あらない → あり＋しない あらない＋は ＝ ありはしない
Your example also contains a contraction. The /w/ has dropped from /ri wa/, which turns it into /ri (y)a/, and this in turn can contract to /rya(a)/:
あらない → あり＋しない あらない＋は ＝ ありはしない → ありゃしない
And the final /ai/ has been replaced with the colloquial pronunciation /ee/:
あらない → あり＋しない あらない＋は ＝ ありはしない → ありゃしない → ありゃしねえ
This is a colloquial and emphatic negative form of ある.