Joyeux anniversaire (Happy Birthday) is usually what you get for your birthday, but today something more royal is applicable. You certainly cannot wish happy birthday to the Queen in the same way you would wish it to your best friend or your child. You can actually celebrate her birthday; you just won’t be able to wish her anything. Well, you can in a certain* way : Vive la Reine! (Long live the Queen) is suitable, especially because you only use the second part of a famous expression.
The complete exclamation le Roi est mort, vive le Roi ! was heard for the first time in France in 1422 at King’s Charles VI funerals, to celebrate Charles VII who was going to succeed him to the throne. “The King is dead, long live the King, says that the King is dead and that we wish a long life to the new King. The juxtaposition* of the two sentences means that the kingdom is not, not even for one second, without a King. We praise here the function and the person who embodies the function.
Since France doesn’t have kings anymore, the expression has adapted to the situation. It is nowadays attached to the country and to its political system, rather than to a person. It is actually used by every French President who traditionally finishes any speech by Vive la République, vive la France. Listen:
Vive stands for que vive which is the present tense of the subjunctive mood. One of the functions of subjunctive is to express wishes, as it is a fact that hasn’t occurred at the time it is enunciated. Subjunctive is usually used for facts that are uncertain; that could happen, but one doesn’t know whether they are going to happen or not. We just wish it! It works the same in English, for example in God save the Queen! (Que Dieu sauve la Reine)
Over time, que happens to be omitted, probably because the expression was first of all used in oral language that tends to shorten sentences.
Also to be noticed that vive has become invariable and so doesn’t have to agree with its subject; it stays in its singular form, whatever follows: vive les mariés! Vive les vacances!
* certain and juxtaposition are words which are similar in French and in English. Slightly different pronunciation though!