I've inherited this old version of ZenPhoto. Recently, logins don't work anymore. Whether I enter the *correct* password or the CAPTCHA (for password reset), it just re-loads the login page. I have made certain that the e-mail address is correct in the database, but I never get a password reset message.

What issues may I run into upgrading from 1.4.0.3 to the latest version? Is there a recommended path that I should take?

That is pretty old and the most secure way will be to upgrade step by step main release version.
http://www.zenphoto.org/news/installation-and-upgrading#upgrading

Up to 1.4.6 that where the 1.x. or 1.x.x releases (since 1.4.7 we changed version numbering a bit - see the releaase post about) that.

There are some issues you can run into like third party/custom theme and/or plugin incompatibilites. As you may imagine hopefully some things changed over the last four years.