asyncio.Task.add_done_callback就是为了在任务完成后（无论成功与否）立即执行某些操作：

import asyncio

from fastapi import FastAPI
import uvicorn

class FooClass:
    def __init__(self):
        self._foo_func_task: asyncio.Task = None

    async def start_foo_func(self):
        self._foo_func_task = asyncio.create_task(self.foo_func())

        # here:
        self._foo_func_task.add_done_callback(
            lambda task: print(f"Exception: {type(task.exception())}")
        )

    async def foo_func(self):
        raise ValueError()

app = FastAPI()

@app.on_event("startup")
async def startup_event():
    app.state.foo = FooClass()
    await app.state.foo.start_foo_func()

if __name__ == "__main__":
    uvicorn.run(app)