When I die will my loved ones be responsible for unpaid bills?

Like phone bills, credit card balances, car insurance etc? Also, does life insurance policies cover suicide?

Yes they will.
Do not do anything stupid. The one thing that you can always do is file for bankruptcy and that will help you out. Call you local Department of Human Services they can help you out.
Life insurance policies do not cover suicide.

Trust me I have been in the same boat. You get all of these phone calls and they make you feel bad. You feel like you can not get put of this debt and you feel bad about yourself. Call a bankruptcy lawyer they can help you out and you can put this past you ans get a new start. That is what I am doing right now.

Unpaid debts come out of the estate. They have to be settled before anything left over is distributed under your will or under the rules of intestacy. You'll have to look carefully at your life insurance policies to see whether there is an exemption clause to cover suicide. Most insurance policies will not pay where a person commits suicide within two years of taking out the policy. Some will pay after that time, the reasoning being that most people who wish to commit suicide will not wait two years to do so.

I very much hope that this is just an academic question.

Life insurance never pays in case of suicide as far as I know.

Only your estate is responsible for paying your unpaid bills. Your loved ones don't have to pay anything with their own money but your savings and investments will be used to pay debt before they can be distributed to your heirs.

No life insurance does not cover suicide. So forget about that idea, go get help.