After lots and lots of work, team of biologists and computer scientists finally made what was deemed to be nearly impossible - a (digital) brain uploading and simulation system. Like every emulator ever, it doesn't recreate brain at physics-level accuracy: that actually already existed for a while, but was so demanding in terms of computations that is was only done as proof-of-concept, well beyond affordable even for the richest.
Instead, brain is simulated based on (complete) understanding of its mechanisms, excluding simulation of cellular and any lower levels, thus making it require much less resources to run. HLE (High Level Emulation) at its finest, if you're familiar with that sort of thing.
In regards to brain activity, main point is that brain is still accumulating memories and impressions. I wrote a more detailed description, but it may be slightly off-topic, so hiding it under the spoiler.
Person keeps interacting with their surroundings, although now also simulated ones. They both can interact with others (of their choice) uploaded in this manner and use various means (mail, chats, video chats and such) to interact with real world. To avoid post-uploading adaptation, all brain inputs and outputs match to what body used to provide.
But this is where question from the title comes in: without cells aging (a nice bonus of not simulating them, I guess) and brain working in theoretically perfect conditions from biological standpoint, will it eventually wear out? While neurological degradation and diseases are often linked to old age, it seems that the longest living human known to science was "mentally sharp" until the end of her life. All in all, 120 years of functioning appears to be well survivable for brain functions, with rest of the body being a limiting factor instead. But without body and even its own cells acting as a limiter, will brain keep working well forever? If not, how long is it likely to take until issues show up?
P.S.: of course, brain uploading is not the only way to waive aging of cells and rest of the body away. One could think of a brain-in-a-jar scenario with brain cells also modified not to count their aging while keeping functioning (and, somehow, fully avoiding cancer), which can be considered the same as HLE-simulation scenario I've described.