I'm an Egyptian citizen, who holds Schengen visa and stay permit in Italy. Can I go to UK for some days with my stay permit and Schengen visa?
The UK is not part of the Schengen agreement. So you would need a seperate visa. The only situation you wouldn't is discussed inthis answer.
The UK is part of the EU, but not part of the Schengen border-free region.
As part of the EU rules on freedom of movement, EU Nationals can travel to the UK without needing a visa (but they do need a passport or National Identify Card to show at the border to show their entitlement to this). Additionally, the Non-EU family members of EU nationals can travel to the UK, but it's a little more complicated. For that case, see the Europa.eu page for details, which explains if the non-EU family member would need to apply for a (free) visa in advance, or if they can travel visa-free.
If you're neither an EU National, nor a family member of one, then the normal UK entry requirements apply to you. There's a handy wizard on the Border Agency site to help you with that one, and when we type in your details (Egyptian National resident in Italy) it says you need a visa.
Being a resident of another EU country isn't enough to travel within the EU visa free, it only generally gives you access to that border area (eg a Schengen, UK-IE common travel area etc). For the full freedom of movement within the EU, it's EU national or family only, sorry!