As a veterinarian, I deal with this question quite often. There is no general answer, it has to do with the individual patient and the client's values. For example, a cat needing a dental evaluation is likely quite painful, because cats frequently get tooth resorption, which is a painful disease. Dental X-rays are very important for diagnosing and treating appropriately, because some teeth that are being resorbed should be extracted ($$) and some can have their crowns amputated (<$). If the client cannot afford to treat, the patient should be monitored to make sure that it has a decent quality of life. Some cats live with pain well, others not so well. This is one of those areas where it is, in my opinion, worse to negotiate middle ground (dental cleaning without x-rays and no extractions) than it is to do nothing.
For mass removals, if the mass is benign, it may not need to be removed. If it is malignant, it should be removed as soon as possible. Some masses have an appearance that suggests they should be removed and sent for biopsy to diagnose them, they just look malignant. Other masses can be diagnosed with less invasive and cheaper means.
When we compare our pets today with our pets from yesteryear, it seems like health is going down the tubes. Our cats have kidney and thyroid disease, our dogs have gingivitis and dental infections, dogs have allergies and cats have diabetes. What is really happening is that veterinarians are able to detect disease that went unnoticed in the past, and they are willing to ask their clients if they want to treat the diseases instead of assuming that "it's just a dog". Our pets are healthier than they have ever been and they stay healthy longer. Cats are living well into their teens, and so everyone seems to have a cat whose kidneys have simply worn out. Young dogs aren't getting hit by cars as often so the gingivitis seems to be turning into periodontitis now. On the other hand, diabetes and allergies are increasing, for the same reasons they are increasing in the human population. The cause of increased diabetes in cats is probably obesity, but we're still not sure why humans and dogs suffer more allergies.
I've rarely seen a pet that needed to be euthanized because of dental disease that the owner couldn't afford to treat.
A good veterinarian will help you figure out where your values are. Personally, I don't think it's good to miss house payments because you had to pay for radiation therapy for a 15 year old cat. But that's not my choice to make for my client. I have to help them figure it out for themselves.