Frame the benefits of user testing to stakeholders in terms they value.
User testing does not have to be a huge new investment
User testing doesn't have to be overly complicated. If you're struggling to find the time, or don't have access to many customers yet, you can always do testing with friends and family. Even asking people on the street or the café in your office building can be a good way to garner opinion and point you in the right direction of needed improvements. Sometimes just five users can provide enough input to alert you to design issues and build an argument for required changes or resources for more solid user testing.
Best practices show that on average, spending just 10% of a design budget on usability is enough to deliver results. That's a cheap way to ensure you spend the remaining 90% correctly, rather than blow your budget on a poorly constructed design.
Of course this might not be the textbook way of doing user tests, but it's better than not doing anything at all.
User testing delivers your organization objective information to aid decision making
When you're heavily involved in a project it's easy for you (and others) to become personally attached to certain features and design elements, but you need to take a step back and look at it with objective eyes.
What's important to you, the developers, product manager, and designers might be totally different from what matters to your end users. When you run your design through user testing you will get an unbiased opinion and perspective on it which will help take the emotion out of important decisions. Incorporating other technology like eye tracking, will give an even deeper level of insight into user interaction. Observing a participant's gaze while using your product or service allows you to virtually see your design through their eyes and pick up on roadblocks or points of confusion you may not know exist. Eye tracking registers users' visual paths and revels how they navigate your design. Unlike click or mouse data, it can highlight points of confusion while reading or navigating as well as other subconscious behaviors that may not come to light during post test interviews.