I believe the issue regarding the propriety of SOS White's actions is pretty clear - he has acted inappropriately and contrary to Illinois law.
Also, I would say that it's clear to me that the U.S. Senate's actions are quite weaselly, even if they are not unconstitutional.
However, the question of whether or not the Senate's actions are actually unconstitutional was a little less clear to me. After having given it a fair amount of thought, I've decided that I think they are. Of course, the right to answer that question with certainty rests only with the federal court system, and in particular, with the Supreme Court. Unless and until they give a specific answer, there really is no precisely correct answer.
My reasons for concluding that the Senate's actions are unconstitutional are two-fold. One of those reasons has to do with SCOTUS' interpretation (in Powell v McCormack) of what rights Congress has to judge the qualifications of would be Congresspersons. Frankly, that argument is not strong enough or clear enough for me to have a definitve opinion based on it; so, I won't bother with it.
The second reason, I believe, is relatively clear. The Constitution gives state legislatures the right to enpower their executive to appoint Senators to fill vacant Senate seats. That is a right that the Constitution reserves to the states - the U.S. Senate has no right to alter or dictate the conditions under which the state legislatures may grant their governors that power. In fact, some states have chosen not to grant that power to their governor at all.
So, the state dictates what must happen for an appointment to be valid and legal. The Senate has no authority to tell them what the state must do to make that appointment valid. The Senate may have its own rules, which contain certain requirements, but those requirements are not necessarilly constitutional. SCOTUS has ruled in the past that they did not have broad powers to establish their own qualifications.
In other words, if Illinois law says that 'if X, Y and Z happen, that means that Person A is the state's legally appointed Senator', then, the U.S. Senate probably has no right to say, 'no he is not because Q didn't happen.' They just can't tell the state what its process has to be.
So, the question is this - according to Illinois law, has whatever needs to happen for Burris to be their appointed Senator happened? Well, the law says what it says, not what someone thinks it should say. I believe the language is plain, all that needs to happen is that the governor appoint someone. No where can I find any wording that suggests that anything else needs to happen in order to make it a valid and complete appointment. If someone can find such wording, I'll be happy to concede the point.
Now, to be clear, Illinois law does place the general duty (*) on the SOS to sign and stamp the paperwork of any governor's actions; and, specifically to register his appointments. However, that has nothing to do with making the appointment valid - it already is. It is something he is required to do, not something that the governor needs him to do to sanction his actions. The SOS simply notarizes the governor's orders, for the purpose of the state's official record keeping. No where does it say that the act is needed to make the actions of the governor effective.
The fact that SOS White has registered the appointment concedes the point, in my opinion. The law doesn't require that he register 'pending appointments' or 'appointments that still need to be certified', only actual appointments. If the appointment hadn't already legally happened, he would not register it.
So, what the Senate is doing, in my opinion, amounts to an unconstitutional subjugation of states' rights. If a state sends them a duly elected or appointed Senator (which they have, in accordance with their laws), they can't say that they will only accept them if they are wearing a blue suit with a yellow tie. To do so, is an assumption of power that they just don't have.
Of course, as I said, it is really up to the courts to interpret the issue for certain.
Just my humble take on the matter.
(*) Note that this duty of the SOS is in a totally different part of Illinois code and has no attachment to the part which grants the governor authority to make appointments. It is a general duty of the SOS for all of the governor's orders. If the SOS's witholding of a signature and stamp makes the appointment not valid, then the SOS has the authority to make every action of the governor invalid. In effect, the governor would need the concurrence of the SOS to do anything - they would be co-governors.