"That heresies should arise, we have the prophesie of Christ..."

It's inversion:

That heresies should arise, we have the prophesie of Christ; but that old ones should be abolished, we hold no prediction.

is a re-ordering of:

We have the prophesie of Christ that heresies should arise; but we hold no prediction that old ones should be abolished.

To paraphrase, I would say, "Christ predicted that there would be new heresies that would arise, so the author can be confidant that they will appear, but He said nothing about when old heresies would vanish, so the author makes no prediction about them." This is in keeping with Browne's observation that supposedly-suppressed heresies seem to pop up elsewhere.

In each independent clause, there is a relative subordinate clause introduced by "that." The first, "That heresies should arise," modifies "the prophesie of Christ" (or stands in apposition to it). The second, "that old ones should be abolished," modifies "prediction."

Both sentences are versions of

It was said that X would happen == That X would happen was said.

The verbal phrase "We have a prophesie that.."
is similar to "We have a law that.. "He made an announcement that...

So,without the inversion and with some "" (round what was foretold) "" and (<< backshift) shown

1 We have the propehsie of Christ that "" heresies should (<< shall) arise,"" ; 2 but... we hold no prediction that ""old ones should (<< will) be abolished.""

In both sentences, the order of the clauses has been inverted. This can be done to shift the focus of the sentence towards the clause that is moved forward.

In a standard construction, I could say:

We have (a prophesy that heresies will arise); but we have no (prediction that old ones will disappear).

By simply moving the italic parts to the front we get:

That heresies will arise, we have a prophesy; but that old ones will disappear we have no prediction.

Note that you might add an of after the verb in both sentences. It is common to add that to refer back to the clause that was moved to the front:

That heresies will arise, we have a prophesy of; but that old ones will disappear we have no prediction of.