If you avoid service a plaintiff can file an affidavit and seek an order to have you served by publication. Then your case hits the legal section of the paper and after a few weeks your are served. But it is not illegal to avoid service.
Just to add to the answer above. After service by publication you will not get notice of actions are taken against you in the court. A judgment can be entered and enforced against you even if you avoid service.
The following should not be taken as legal advice, but simply as information based on general principles of law which is intended to educate. If you need legal advice, you should consult a lawyer.
It is not illegal to avoid service of process. In most cases, however, service is inevitable. As Mr. Alexander notes, if you avoid service long enough, the plaintiff may still serve you by publication in most cases. If you are being sued and are liable for attorney fees and costs under a contract or statute, the fees and costs of publication will be added to the judgment that you might eventually have to pay. Avoiding service, such as not answering the door or coming out of your office to be served, is one thing. Evading service, such as lying to or running away from a process server, is another. Evading service is not illegal either, but it could give the judge a first impression of you as being dishonest or evasive, and could be used against you in court later.