To simplify the situation, consider a team of 4 people - Bob, Jake, Dan and me.
There are 2 distinct areas of responsibility in which our team operates. One that belonged to Bob and Jake, and one that belongs to Dan and me.
Recently, Jake was let go. This was an upper management decision. Bob was already pushing deadlines due to the amount of work. Jake, although less experienced, took some of it from him. Now Bob is alone and he certainly won't cope with basically twice the work. There are no plans to hire Jake's replacement.
Our direct manager knows that as well. What I was told by him is that I am to take over Jake's responsibilities. Trouble is, I'm in the same situation as Bob - I already am "full" on my own responsibilities, and to top it off, I have no experience in Bob's and Jake's domain. I also cannot delegate the work to Dan as
a) The manager told me not to
b) The manager told me he doesn't believe Dan has enough experience to handle it.
It seems quite contradictory to me, as I also have no experience in that regard. True, I always like to learn new things (and show it in the workplace), but I really don't see how that's going to work.
I expect answers about asking my boss to prioritize - I did that, and it's basically "everything is important". Due to the nature of the tasks (Bob's domain is more everyday tasks, while mine is long-term projects), I am pushed to do more of "not mine" work as it produces results and looks nice on the KPI boards.
My question is: what other options do I have to show my boss (or his boss) that the current situation would lead to unfavourable outcomes? Bob already told me he is considering to change jobs, and I am as well, considering I do not do what I was hired to do and I'm afraid of "going out of the loop" with my skills.
EDIT: To address some points in the answers and comments.
Overtime - Overtime is actually actively discouraged. I am supposed to complete the tasks in a normal, 40 hour workweek.
Company too low on money to hire Jake's replacement - I don't believe it's the issue of money. This is one of the largest corporations in the world in that product domain. Unofficially, I heard the upper management reasoning was:
Hey 2 years ago Bob was alone and things went pretty smooth, so Jake isn't needed. This however misses the fact that over the last 2 years the workplace has grown in size (both spatially and employee wise) by at least 250%. So the workload from 2 years ago is nothing compared to the workload today.
Keeping the lights on vs long-term projects - The long term projects I'm responsible for are actually also keeping the lights on - whole parts of the company depend on their continuous development and maintenance. If I stop doing MyTaskA to do JakeTaskA, then one of our departments is, well, SOL on necessary changes.
Difference on tasks - when I say I'm not familiar with Jake's tasks, I mean I know nothing about them, have no experience and never done something like them. It's broadly the same domain, in the same sense that fixing an engine at your local garage and designing one in the R&D labs are the same - hey, it's working with engines, right?