The warning you are getting are timing violations. You get these because the tools detect that your design does not obey the necessary timing restrictions for the internal primitives.
For instance, inputs to lookup-tables (which is one of the main building-blocks inside an FPGA) need to be held for a specific time for the output to stabilize. This is very hard to guarantee when your entire timing relies only on the latencies and delays of the components themselves, and switch on a completely asynchronous basis.
Depending on your actual design (mostly the size and complexity of it), I'll wager the guess that you'll end up with a lot of very-hard-to-debug errors once you get it inside an FPGA. You'll have a much, much, much easier time using a clock. This will allow you to have a clear idea of when signals arrive where, and it will allow you to use the internal tools to check your timing. You'll also find it much easier to interface to other devices, and your system will be less susceptible to noisy inputs.
So all in all, use a clock. You (probably) wont damage your FPGA by not doing it, but a clock will save you from tons of trouble.