If I build a website and don't add screen reader tags to any of the content is that discrimination or illegal in anyway? Is it possible someone with assistive needs could potentially sue the developer for discrimination?
Yes, it's possible that you (the developer) and/or the owner of an inaccessible website could be sued if the website isn't accessible.
The Americans with Disabilities Act has recently been used in a lawsuit against Winn-Dixie. Their website wasn't accessible to blind users, and the ruling was that Winn-Dixie has to update their website. This is the first case where the ADA has been used in this way, and in this case, it was applied because Winn-Dixie has bricks-and-mortar stores the website is heavily integrated with. So, the ADA most likely won't apply to standalone websites, but there is also Section 508, which currently applies to federal agencies and those who receive federal funding.
In summary, it's possible but unlikely that you or the site owner could be sued, but it's certainly best practice to, at minimum, inform the site owner/client that there are some laws around the issue and they'll be excluding some visitors if they don't allow you to spend some time making sure the site is accessible. There are many things that you, as a developer, can do to ensure the site is as accessible as possible - for example, make sure visitors can use just a keyboard to navigate throughout the site, include alt text and form labels, and write clean, semantic code - without even billing extra specifically "to make the site accessible."