Now that I have experience in my current field working for someone else, when should I consider removing the old positions from my resume?
The key point you need to focus on in order to understand this question is that a resume is a sales tool. You're selling yourself to potential employers. It's just like an advertisement in the newspaper. When Apple takes out a full page add in the New York Times to announce a new iPhone, they don't also include photos of the PCs they were making in the 90's. Instead, they focus on what makes the new iPhone relevant and interesting for it's intended audience.
You need to do the same with your resume. First, you need to understand your audience - and then you need to target your resume based on what they're looking for. Consider that employers are generally looking for two things (or, two categories of things) when reading a resume:
Is this person skilled, knowledgeable, and experienced in the field I'm hiring for? In other words, if I'm hiring for a C++ developer, does this person have skills in writing C++ code? This is the easy and most obvious resume feature to write for - if you're looking for a job in line of work B, and you have recent experience in B that has given you valuable skills, you need to make sure that is a focus of your resume - the jobs where you learned those skills should be prominently featured and you should give concrete examples of work you've done, or other specific information to show your skills.
Is this person a good employee? Employers are concerned about how you will perform in their work environment - will you be able to communicate with your peers and clients? Can you interpret fuzzy requirements and ask good questions to clarify them? Can you follow the rules? Will you slack off all day once a project has lost your interest? Will you quit after 3 weeks because you got a different offer somewhere else? This can be a much harder thing for people to show off on paper, since you can't just write "I'm a good employee." It's also more of a risk evaluation for the employer, essentially they're trying to feel out if you'll cause problems or not: even the most skilled employee can upset customers or job hop or whatever.
Your question really focuses on the second bullet point. If I could rephrase what you asked, it would be:
What value, if any, do old positions in another line of work add to my resume, now that I've had a few years of experience in a new line of work?
It sounds like your skills at "A" are not really relevant, but your work history as an employer may be. At the very least, showing you have years of employment can be helpful to show you have some continuity. It can also give you a starting point for "soft skills" questions you may be asked in the interview. That said, it does sound like you had a bit of a rocky timeline, so you may want to keep it simple:
Various employers and positions, 2009 - 2014
Responsible for web application development
This way, you show that you were gainfully employed, and you have a point of reference to discuss what you'd been doing in the past. Interviewers who see this may ask why you switched careers - so make sure you have a positive answer that focuses on why you like B and what you hope to accomplish there. In other words, take advantage of showing off that you made a deliberate switch to B, if possible.