Technology

Replying to you, but it is valid also for @[email protected].

If you ask for permission to do certain works in your house, you present the project to your city council, or the required office, and if after a given time (depending on what what you want to do) they don't object then you have the permission. Before the introduction of the silent consent, you have no idea about how many time you need to wait before you get an answer and it was prone to corruption while now the "yes" is the default unless there are real problems. It is not a perfect solution, but it is way better than before.
Basically all the interactions with the authorities are on a silent consent base when the authority in question does not need to produce something to give back.

All the minor changes to the contract with banks, utility companies and so on: they propose the new terms and if you don't accept in a given time from the moment you read it you accept it. By law in the event I refuse the new terms, I don't end with the old ones but the contract end and in the case it has penalties for early terminations, these are nullified if the penalties are applied to the other side.
On the other hand, this way a company has a certain deadline after which the new terms come into effect and as a side bonus the fact that it has to handle only the exceptions (who don't accept) and not all the ones that are ok.

Wedding publications, since we have not the whole "if you disagree to this marriage talk now or shut up forever" part of the ceremony, to be sure that there is no hidden problems we put an announce in a designated public place (usually a notice board at the town hall and/or your church) for a given period of time, usually 2 or 3 weeks, and then if nobody object you can marry.
I agree that this is probably something old that were done back at the time but it work on the same principle. Of course now there are other ways to know if someone is already married (on the civil side) or is divorced (on the religious side) or there are some hindrances.

And before someone ask, we also have examples where this approach were shoot down: the last of these is when a big back decide to move part of their clients to a virtual back (a different branch of itself) and they were stopped on the basis that this change it too radical to be done this way (even if the notice was about 6 months). Other cases hit utilities companies which in some cases where forced by a judge to pay compensation to the customers because what they done was basically illegal and the silent consent where then void.

Not OP, here's a Law Stack Exchange thread where silence (in conjunction with other actions) is consent.
Here's where silence is NOT consent.