When this happens, will the Mono framework drop Web Forms too, or will they continue to develop?
The short answer is yes, they will drop Web Forms too. You should consider Web Forms deprecated—both in terms of .NET 5 and the forthcoming versions of the Mono runtime. You will continue to be able to run Web Forms on legacy version of Mono and .NET Framework—at least insofar as you're able to today—but this support won't be carried into future versions.
To better understand this, it's useful to understand the relationship between .NET 5, the CoreBCL, the CoreCLR, and Mono. This has become a bit fuzzy since the CoreCLR now works across multiple platforms, thus overlapping with much of the territory previously handled by Mono.
At a high level, .NET 5 will utilize a newly unified .NET Core Base Class Library (or .NET Core BCL). That BCL will replace many of the class libraries that were previously handled by the various .NET implementations, including Mono. Mono, however, will continue to be used as the runtime for .NET 5 in environments where size is a higher priority than performance, such as mobile and WebAssembly. By contrast, the faster (but much larger) CoreCLR will (continue to) be used in other other environments, such as desktop and web server applications. The .NET Core BCL will operate on top of these runtimes.
In other words, in .NET 5, there won't be separate class libraries for Mono, as there is today. Everything will use the same base class library. And, on top of that, you'll have additional class libraries for e.g. ASP.NET, WPF, Xamarin, &c. That will be completely independent of what runtime they're using. As a result, since ASP.NET Core 5 won't support Web Forms, that means that Mono won't either, since they'll now be using the exact same class libraries.
There's an excellent interview with Miguel de Icaza on InfoQ that is worth reading for better understanding the direction of the Mono runtime and how it relates to .NET 5. (Miguel is the original author of Mono, co-founder of Xamarin, and continues to direct Mono development.) It doesn't talk about Web Forms specifically, but does a good job of clarifying the relationship between Mono and .NET 5.