It really depends on the company. I've been instantly approved with a score of 543 (Target for the red card), with a 636 (for a prime card with a high limit), with a 602 (for a prime card, albeit with a low limit). In the latter two cases, it was more based on internal history with the company.
I'd say in general, 720+ with no major delinquencies (e.g. no public records, no collections, no chargeoffs, nothing ever 90+ days late) is likely to be instant.
Note that some banks don't do instant approvals. Chase has only a few cards with instant approval. Juniper doesn't usually do instant approval for second or subsequent cards.