There are likely many different ways people deal with this. I think it would depend, to some extent, on the personality of the character in question. Are they the kind of person who likes to deal with things right away, or are they someone who hates visual reminders of sadness? Or are they the kind of person who is more sentimental about things if those things are associated with something or someone they love? Some people need to talk about and think about the person they've lost, and the memories make them feel better, as if the person still isn't too far away. Others get even sadder (or angry) if someone even mentions their loss or wants to talk about the person who is no longer there (I've run across friends on FB, for instance, who have asked for fond memories to not be shared on their wall).
I've read books where a protagonist who had lost a child left the room as sort of a frozen-in-time shrine, for years, and others where someone left their lost child's room closed up and unchanged because they couldn't stand to go in and clean it out. I suspect some people would wait a while, maybe even years, before cleaning it. Others might do so sooner, maybe even right away, in an attempt to remove the constant reminder of their loss. When my dad died, my mom waited about a year, then cleaned out his closets and the garage (his workshop) and gave the things she couldn't use and didn't want to keep to charities. She left his model airplanes are still hanging up in his old bedroom, though. It's possible a parent might keep a few things that especially remind them of their child in that way.
I'm not exactly sure what I'd do personally, not having children. Given my penchant for procrastination and for holding on to things that have memories for me, I'd probably wait a long time, maybe even years, to deal with my lost child's possessions and room. Unless someone really pressured me to deal with it sooner.
Note: Brightdreamer makes a good point too; the circumstances of the death could also have an effect.