First of all, you can travel freely with your son - there are absolutely no restrictions; neither in Dubai nor anywhere - that you must be married to the biological parent of your child in order to travel with him/her.
If you are not the biological parent or not the legal guardian, then you must carry some documentation that gives you authority to travel with the minor and act as their ward; but this is not the case with you.
Next, the issue is how are you getting into Dubai? If your fiancé has applied for your visa as a spouse then he must have shown a notarized marriage certificate.
If he is sponsoring you as a family friend then no worries at all.
If you don't need a visa (for example, you are eligible for visa free entry or are getting a transit visa or a visa on arrival), you have nothing to worry about.
Finally - Dubai police will not take away your son if the child is born out of wed lock. This is only done in very extreme and rare cases (for example, in the case of rape or abuse); and frankly I have yet to hear of this done in any GCC country (even conservative Saudi Arabia).
The only problem you have is that it is illegal for opposite sex couples to cohabitate if they are not married or otherwise related. This only comes into play if you are reported to the police; and you will only be reported to the police if you give reason for people to do so.
You may also face this hurdle if you are staying with your fiancé at a hotel; some might ask for a marriage certificate - especially if you are rowdy and give them cause to do so.
Dubai police are not going around knocking on people's doors checking marriage certificates.
If you plan to move to Dubai and live with your fiancé, then you'll have some concerns as without a marriage certificate, you and your son cannot be sponsored by your fiancé/husband - which means you cannot enjoy freedom of movement in and out of UAE, have your son enroll in school, get a driving license, open a bank account and all the sundry that one does when moving to a new country.
However, for a short visit - not a problem at all. Relax, and enjoy your trip. Depending on how old your son is, make sure you plan a visit to LEGOLAND =)