Your employee is right to ask for more money to take on more responsibilities and take more of their time.
A comment on PeteCon's answer puts it succinctly.
That's why there isn't really a labour shortage, it's actually a payment shortage. – Peter M
The idea that "overtime exempt" employees (US jurisdiction) are basically constantly available is an abuse of that classification.
Edit: I had a good link for violations, but included an irrelevant quote from it. I'm replacing it with a better source.
1. Believing Salaried Employees Are Not Entitled To Overtime
Many employers are under the mistaken belief that only employees earning wages on an hourly basis are entitled to overtime. The fact is that an employee is not prohibited from receiving overtime wages simply because they are paid a set salary. In order for a salaried employee to be prevented from receiving overtime, the employee must qualify for one of the specific exemptions provided by the Fair Labor Standards Act (“FLSA”). ...
8. On-Call Work
At times, employers require employees to be “on-call” when they are not scheduled to work, requiring the employee to be available to be called into work at a moment’s notice. When this occurs, employers often fail to pay the employee for this time. Determining whether an employee’s on-call time is considered work-time for which compensation is required depends on the facts involved. The general rule is that if an employee is required to remain on-call on the employer’s premises or so close that they cannot use the time effectively for their own purposes, the waiting time is considered hours worked under the FLSA and is compensable.
The rest of that page is an excellent resource for plenty of very common violations, so everyone should read it to learn the law to protect yourself from those abuses, too.
And just so we're clear on what "overtime exempt" classification entails, here's another article to read. Here's a the highlights.
First, the government looks at the employee’s annual pay. Workers who make less than $23,600 per year – or $455 per week – are automatically considered non-exempt employees. ...
Next, the government will determine whether the employee is paid on a “salary basis.” Essentially, this means that the employee is guaranteed to make a set amount of money in any given week that some work is performed. ...
If the first two tests are met, the government will then look to see whether the employee performs exempt job duties. Only executive, professional or administrative duties will be considered exempt. ... The professional duties exemption is meant to cover what the DOL refers to as “learned professions” including doctors and nurses, lawyers, architects, clergy, scientists, accountants, engineers and other jobs that require advanced knowledge. ...
If all three of these tests are met, the employee can be properly classified as exempt. If not, the employee will be considered to be non-exempt and must be paid at least 1.5 times his or her normal hourly rate for any overtime worked. Overtime is defined as working more than 40 hours during any given workweek. ...
(Bolding for emphasis is my edit.)
These abuses aren't just something new, either. They've been happening for decades, as this 1999 article describes as happening for years even then. And it includes a section specifically for software developers:
“For example, California doesn’t have a specific exemption for computer professionals, which is available under the federal law. [So] California employers should treat any programmers who spend more than 50 percent of their time writing code as nonexempt.”
- Mae Lon Ding, president of Anaheim Hills, California-based Personnel Systems Associates.
Abuses are so common (and unreported*) in the US that I wrote a calculator to show how much money employees are losing when they work unpaid overtime, trying to educate workers on how much they are really are missing out on. This calculator includes the equivalent pay someone working hourly would be making if they were doing the same job and getting paid the legal minimum overtime rate. It also calculates how much employers are saving by just in payroll for not paying overtime rates.
In the default example in the calculator above, an employee working an average of 45 hours a week at $20 an hour is losing $7800 gross yearly pay by not getting paid for their overtime. An experienced software developer making $50 an hour at 45 hours per week is missing out on $19,500 per year. Of course, working more hours and making more money makes the problem even worse.
Oh, you think that's an exaggeration? Well, working a single hour per week without getting paid for it at $50 an hour is $3900 in lost wages for an employee.
And you want your employee to do more work and work longer hours without a pay raise? And you want them to possibly miss sleep, have to miss their kids' school events, interrupt private time with their spouse/significant other, or 1000 other situations in their private life?
Yeah, that's probably not going to last very long, even for the employees that agreed to it. And that will depend also on how often they get called after hours. If it's common, then you can bet it will be pretty quick that you have the whole department asking for a raise.
Personally, I won't work overtime or even take a job that requires on-call rotations. There's no benefits for me, since there are other jobs that pay the same (or higher) for positions that don't require it.
* Most cases are unreported because the employee doesn't know what the laws are.