If you're getting deprecated errors that you didn't see before (which IMHO you shouldn't be seeing on any pages of a live site, only in your error logs), it means your version of PHP has been updated. Deprecated means that a "style" of coding that was OK in an earlier version will at some point cease to work. The deprecated errors give you a chance to fix things before another version comes out that will no longer support that style of coding.
I'm not too much into OOP (years and years of procedural have hindered me a bit I'm afraid), but "new" suggests an object. "reference" means the actual object and not a "copy" of it.
I get lots of these errors in my WordPress blog. Why? Because WordPress wants to maintain support for PHP version 4. And as the code still "works" in PHP version 5, I guess they're in no rush to fix things quite yet.
If the errors are coming from 3rd party files, or if it's your code and it might be used by others using PHP 4 you may not have much choice but to live with them for now.
But if it's your code and you're certain you won't be using PHP version 4 again, IMHO you should make an effort to fix the errors.
I would be very interested in seeing an example line of code that's throwing the error and how others would change things.