Is there a service that would inform me whenever a new direct route is scheduled from a given airport?

As a frequent traveller and maintainer of OpenFlights, I'd love it if there was such a thing, but I'm not aware of any such global service and up-to-date route/schedule data in general is pretty tightly locked down with copyrights.

What I do instead is subscribe to relevant blogs and news sites. Some random examples:

• Australian Business Traveller keeps a close eye on flights to/from Australia.
• Routes Online is a firehose of global updates, with tags that you can use to keep an eye on individual airlines and airports (example).
• Most airports and regional aviation authorities also have news feeds that contain updates about new routes; for example, here's Finavia's news for Finland.

Since you're interested in specific city the number of airports you're considering seems limited. You may check if they have some sort of a newsfeed/rss on the airport page. I may be wrong, but airports like to brag about any new routes they managed to attract to the airport so the info should be on their webpage. Check if they don't have a newsletter to sign up for.

Even if the answer is no, still all airports I have ever checked have a page for departures/arrivals schedule (not the next x, but the plan for the whole week). Use a tool to track changes on such pages. As mentioned in a comment to a different answer, there are browser add-ons (I used to use UpdateScanner for Firefox, I don't know if it is still working though) to do that.

Another option you may consider is looking for some newspapers regarding airliners or aviation. If there is anything local enough, they may be listing this kind of information (according to their publishing schedule, so probably once a month).

Either case you'll also get some extra junk but that you can rule out by using filters/rules in your mailbox. Changes tracking on the departures page is probably least vulnerable to false positives (i.e. you're notified when there's actually nothing interesting changed).

One (not ideal) solution is to look at the list of the main airlines serving your airport, then look at these airlines' webpage on their new routes.

E.g., for SFO, https://www.flightconnections.com/flights-from-san-francisco-sfo mentions that:

Airlines flying from San Francisco. In total there are 44 airlines flying from and to San Francisco. Most flights are operated by one of the following carriers:

• United Airlines (UA): 109 destinations
• Alaska (AS): 37 destinations
• Delta (DL): 9 destinations
• American Airlines (AA): 8 destinations
• Southwest Airlines (WN): 8 destinations
• [...]

Then look at these airlines' webpages pertaining to their new routes.

• https://www.united.com/en-us/new-routes (mirror): new flight from/to Bangalore (BLR).
• https://www.alaskaair.com/content/new-flights (mirror): new flights from/to Anchorage (ANC), Bozeman (BZN), Cancun (CUN), Ixtapa-Zihuatanejo (ZIH), Jackson Hole (JAC), Loreto (LTO), Mazatlan (MZT), Missoula (MSO).

However, as mentioned in another answer, the Wikipedia seems to be quite well updated, e.g. https://en.wikipedia.org/wiki/San_Francisco_International_Airport#Passenger