The colour red is usually associated with "danger" and "you're about to do something dangerous." What should you look out for if you're planning on a design that is primarily focused around the colour red? Also, in that case, is red still a "correct" colour for warnings?
@DA1 has pretty much covered the main aspect that it would depend on the context i.e. a red call to action to button might stand out more strongly and convince people to click it as opposed to one with a lighter shade.
His second point about branding is also spot on.
Another aspect I would recommend that you consider is that you should also look at the meaning of red from a cultural/regional perspective.What might be used as an indication of danger or evil in some cultures might actually have the exact opposite effect in other cultures
10 Most Commonly Used Colors In Web Design And Their Examples - Interesting article on different colors being used and their impact from a design and emotional perspective
Red is often associated with alerts/warnings, but it doesn't have to exclusively be associated with that. If your site is primarily red in theme, then you can certainly pick out other colors to serve the purpose of alerting users.
Take a look at Target's web site. Red is their primary branding tool and, as such, a huge part of their site's UI. They then use other colors as their secondary emphasis colors (such as the sky blue for product links).
It much depends on the context and the surrounding colors. Here's a comparison of two red buttons, in very different surroundings. Like DA01 already commented, red doesn't exclusively need to be associated with alerts. Some other examples: ColoCola red, red light district, red roses, etc.