Photoshop's Save for Web ability contains a setting called Convert to sRGB. If on, it destructively changes the resulting file's colour values from the document's profile to sRGB. I believe this is the wrong thing to do in almost every conceivable scenario. The default behaviour is for Convert to sRGB to be enabled. I think this is a huge mistake.
Let's take a look at some common scenarios.
You're building a website using GIFs:
GIFs can't contain ICC profiles. This means if you're using GIFs, they can't benefit from any colour management at all. Converting to sRGB when saving for web will destructively change their appearance with no benefit. If you've used a specific colour, like #FF0010 in your GIF, it will likely be changed to not match the same colour used in HTML, CSS or other code.
You're building a website using PNGs:
PNGs can contain ICC profiles, but PNGs saved using Save For Web can not. This means if you're using PNGs and Save For Web, they can't benefit from any colour management at all. Converting to sRGB when saving for web will destructively change their appearance with no benefit. If you've used a specific colour, like #FF0010 in your PNG, it will likely be changed to not match the same colour used in HTML, CSS or other code.
You're using a PNG or JPEG image with a colour profile on the web, and it's being shown in a colour managed browser:
In this situation, you wouldn't want Convert to sRGB turned on, you'd want to store the document's ICC profile within the image and let the browser do a realtime correction, based on the viewer's computer and settings.
You're building an iOS app:
iOS uses PNG files almost exclusively for app design. I believe iOS ignores ICC profiles stored in PNGs (this is smart for many reasons, including performance). The best way to ensure colours look good on the device is previewing your UI on the device itself. There is some variation between iOS devices, but Convert to sRGB does not improve consistency. Converting to sRGB when saving iOS assets will destructively change their appearance with no benefit.
You're building an Android app:
Android uses PNG files almost exclusively for app design. I believe Android ignores ICC profiles stored in PNGs (this is smart for many reasons, including performance). The best way to ensure colours look good on the device is previewing your UI on the device itself. There is wild variation between Android devices, but Convert to sRGB does not improve consistency. Converting to sRGB when saving iOS assets will destructively change their appearance with no benefit.
I can not think of any scenario where Convert to sRGB makes sense. If, in the highly unlikely event you do need to convert a document to sRGB, it can be done using Edit → Convert to Profile.
Why was Convert to sRGB added in the first place?
Further reading.
Colour management and UI design — My article. Explains the situation with more depth.
A search for "save for web color shift" returns over 8 million results. This is a real and significant issue for many people. An issue that can be fixed by changing a single default setting.
Save For Web, Simply — Note that the settings recommended match what I'm suggesting.
This is a big deal.
This has been demonstrably incorrect for a very long time. I don't know any web or app designer worth their salt who keeps Convert to RGB on. In fact, I'd go as far as saying that if it was permanently turned on, I wouldn't be using Photoshop for any screen design work.
By changing the default behaviour, I think Adobe could remove a lot of frustration for Photoshop and Illustrator users.