If the market is any indication, the answer is a resounding no! Check out market share research conducted earlier this year and published on Silicon Angle. The two largest commercial application servers Websphere and Weblogic have a whopping 2.17% market share between them. The lion share of the market is going to open source application servers such as Tomcat and JBoss.
As an architect and developer, I've always thought open source application servers easier to support. In most organizations, access to software vendor support staff is tightly controlled and requires a bureaucratic effort to utilize. Because of this, developers are often left attempting to resolve issues on their own anyway.
Self serve resources for commercial application servers such as documentation, web postings for similar problems, and searchable bug lists are often not much better than what you find with the open source alternatives. Furthermore, getting to knowledgeable support staff often takes time I don't usually have. Often, my support calls need to get routed from first-level support to second or third level support.
Most problems that start out being blamed as "application server" issues are usually application code defects. Rarely are outage or production defects resolved at the application server level. This makes sense as application server code is usually better and more thoroughly tested than application code.
As a manager, I've never found the "security" of having vendor resources available particularly comforting. It certainly doesn't assuage clients who are experiencing some type of outage or defect they need help with for very long.
Centralize and standardize use of sophisticated software (whether its commercial or open source) throughout the enterprise. Standardize the use of application server software to the point where all the typical issues are solved once and do not need to be continually revisited. For example for application servers, I standardize all build and deployment scripts and container configurations. With the exception of memory allocation and port assignments, I standardize other feature usage (e.g. management console usage) so that they are the same for all deployed applications.
These choices are not usually revisited by each application developer or team for each application. When the need for changes arise, these changes are centrally evaluated and those configuration standards changed. The change is then deployed on a planned basis throughout the enterprise. This may seem a bit excessive, but I'd rather developers spend time adding needed software features to applications and better supporting the business rather than low level application server configuration concerns. As a result of this standardization, mysterious problems occurring in some environments and not others rarely happen. I've written more about the benefits of this type of standardization here.
Some organizations see liability benefits to commercial software. For instance, it's another firm to possibly to shift blame to should problems and issues arise. Maybe it's where I've worked in the past, but I've never seen blame shifting strategies of this type work over the long term.
Another inference from this market share study is that use of Enterprise Java Beans (EJBs) has largely disappeared. However, that should be a separate discussion.