The first thing is I would check that the emails settings are correct and you have logging turned on so you can debug the issue - https://stackoverflow.com/a/10810193/1410540
I would then check your email template for the Forgot Password and make sure there are no errors in the template. This template is set under System -> Configuration -> Customer Configuration -> Password Options -> Forgot Email Template.
The next thing to check is if emails are being sent on your server. You can check this a multiple different ways:
Send a email through the contact page. Make sure your email is set as the receiver of the email under System -> Configuration -> Contacts -> Sends Emails To.
Place an order as a guest/new user to a different email address and see do you receive an email to that email address. If you are using Magento 1.9 you will need to make sure a CRON is running as sales order emails are sent from a CRON in 1.9.
At this point if you are not receiving emails I would check to see if any exceptions are being thrown in the logs or use a module such as Magento SMTP Pro Email Extension - https://github.com/aschroder/Magento-SMTP-Pro-Email-Extension to debug the issue.
However if you did receive an email from either scenario above I would disable all community and local extensions (move the related XML files under app/etc/modules into another directory temporarily) and see if you can receive the email at that point. If you did then start adding back the extensions until you find the extension causing an issue.
I hope this helps.