I want to cut the crossbar on a mens bike to make it easier to get on and off. Will it damage the bike?
This is downright dangerous and should never be done on a bicycle someone intends to ride. The top tube is integral to the strength of the bike. The frame can buckle or worse when you try to ride it.
First, make sure you're looking at the right size of frame. If its too big, it's going to be hard to get on and off from. You can buy a stepthrough frame or a mixte frame if you want something easier to get off and on, or a recumbent bike.
When you sit on a bike with a traditional diamond frame, your weight on the saddle pushes the bike down and this force is resisted by the wheels where they contact the road. The tendency of the bike to "sag" in the middle under your weight is resisted by the top tube – without it the seat and handlebars would be forced towards each other by the reaction of the wheels to your weight. I'm not sure if an adult could even sit on a bike with the top tube removed, there would be a large bending load on the downtube.
Frames that are built without a traditional top tube (for example mixte and step-through frames) use other means to resist this tendency and are not (generally speaking) able to be as strong or as light as the diamond frame. One of the very wonderful things about bicycles is the elegance of design that goes into creating light, strong structures. Every bit has a purpose.
I've often wondered this myself. There is a manufacturer that eliminated the downtube and replaced it with a cable.
I have broken the seat tube on a frame and ridden it home that way. It was more flexy, but it was rideable.
When you ask these kinds of questions, you get an immediate kind of response exactly as you see above, usually with no empirical basis.
Based on what it felt like to ride a bike with broken seat tube, I think you'd find WAY more flex, and as the frame flexed, your seat tube and headtube angles would change. I believe you'd notice that in the way the bike handles. Would the frame eventually fail? It probably depends on the frame. If you have a chunky steel frame, it might just flex and steer differently, but hold up indefinitely.
If it's a steel bike you can afford to lose, my advice would be to try it and see. Steel fails in a slow predictable way. You'll see if it bends past its ability to recover or start to crack. Take it on an easy ride first, somewhere flat.