.sr-only is a class name specifically used for screen readers. You can use any class name, but .sr-only is pretty commonly used. If you don't care about developing with compliance in mind, then it can be removed. It will not affect UI in any way if removed because the CSS for this class is not visible to desktop and mobile device browsers.
There seems to be some information missing here about the use of .sr-only to explain its purpose and being for screen readers. First and foremost, it is very important to always keep impaired users in mind. Impairment is the purpose of 508 compliance: https://www.section508.gov/, and it is great that bootstrap takes this into consideration. However, the use of .sr-only is not all that needs to be taken into consideration for 508 compliance. You have the use of color, size of fonts, accessibility via navigation, descriptors, use of aria and so much more.
But as for .sr-only - what does the CSS actually do? There are several slightly different variants of the CSS used for .sr-only. One of the few I use is below:
margin: -1px 0 0 -1px;
The above CSS hides content in desktop and mobile browsers wrapped with this class, but is seen by a screen reader like JAWS: http://www.freedomscientific.com/Products/Blindness/JAWS. Example markup is as follows:
<a href="#" target="_blank">
Click to Open Site
<span class="sr-only">This is an external link</span>
Additionally, if a DOM element has a width and height of 0, the element is not seen by the DOM. This is why the above CSS uses width: 1px; height: 1px;. By using display: none and setting your CSS to height: 0 and width: 0, the element is not seen by the DOM and is thus problematic. The above CSS using width: 1px; height: 1px; is not all you do to make the content invisible to desktop and mobile browsers (without overflow: hidden, your content would still show on the screen), and visible to screen readers. Hiding the content from desktop and mobile browsers is done by adding an offset from width: 1px and height: 1px previously mentioned by using:
margin: -1px 0 0 -1px;
Lastly, to have a very good idea of what a screen reader sees and relays to its impaired user, turn off page styling for your browser. For Firefox, you can do this by going to:
View > Page Style > No Style
I hope the information I provided here is of further use to someone in addition to the other responses.