I don't think they would have had a significant impact if they had survived. The glory days of the monastic warrior orders was over. There were not any Crusader States really left to protect.
The Teutonic Knights had already developed into a order state that functioned more like an actual state than a military order. When the Reformation came, the order secularized and became Prussia.
The Hospitallers retained a fighting role because of their position on Rhodes, but essentially became pirates against Ottoman shipping and helping the dominant Catholic naval powers in the Mediterranean. After the Siege of Malta and Battle of Lepanto in the 16th Century, they became increasingly insignificant. They survive today as merely a Catholic order engaged in charity work.
The Templars have neither an significant contiguous land area they can call their own, nor a frontline presence against the Ottomans. They will probably be interested in forming some kind of monastic state, but I am not sure where that would be located. In any event, the local nobles would be against it.
I see four possibilities.
1) They remain essentially an international banking network and secularize sometime, probably in the 17th to 18th Centuries. Some kind of Catholic order probably remains distinct from its bank.
2) They form the basis of an international militant Catholic fighting order against the Protestants during the Wars of Religions using their membership and wealth to support Catholic kings against Protestant leagues and such. Their importance fades once some kind of civil peace is achieved. Eventually, any business aspects become secular although there may continue to be some kind of Catholic order.
3) The Templars succeed at forming some kind of monastic state. Unless this is in some kind of Northern European land where Protestantism takes hold, it remains Catholic. Eventually the order state secularizes and is likely taken over by some major power. Possibly, the order may find such land in the New World under the auspices of Spain or France (probably some small island in the Caribbean).
4) The Templars survive for a while until another king levels trumps up charges against them and destroys them.
In either case, their continue survival becomes an interesting footnote, but probably does not change history much.