I think that there would be little practical impact, except for those people who are heavily invested in DTC DUATS specifically and have to switch to another system.
First, there are two DUATS providers: DTC and CSC. Even if DTC goes away, CSC is still available, at least for now, and it provides exactly the same service as DTC (see this questionthis question).
Second, Lockheed Martin Flight Service (LMFS) provides all of the information and flight plan filings that DUATS does, and also provides briefings with a human briefer. So there is a completely separate alternative to DUATS available, although admittedly switching tools always has some cost in terms of time and learning at least.
I assume the main impact would be on people who use DTC DUATS as their primary flight planning and filing tool and use it directly, rather than via another tool. For example, I use Garmin Pilot with CSC DUATS but I could switch to DTC simply by changing the data provider option. Some flight planning tools also support LMFS flight plan filing and I guess that more and more will in future, if only for competitive reasons. But if you use the DTC DUATS website directly and are very experienced and comfortable with it then switching to LMFS or anything else would have a learning curve.
But in the end, people won't quit flying or even cancel flights just because DTC goes away. They'll just complain a bit, choose a new platform (probably LMFS) and get on with it.