In general, after denaturation velocity of protein in gel-electrophoresis should change due to loss of the native conformation. Loading buffer for SDS-PAGE usually contais beta-mercaptoethanol, reducing S-S bonds, and SDS. The latter coats protein molecules, masking their own charged groups, leading to overall negative charge of protein molecules. Thus different protein with wide range of intrinsic charge became negatively charged; the amount of charge acquired by certain molecule directly correlates with its size.
in principle if the protein is monomeric in the folded state, a completelly unfolded protein run slowly in SDS-page respect the same protein in a globular folded state (which is more compact and it is characterized by smaller hydrodinamic radius)
However, if in SDS pageif you can really distinguish beetween folded and denatured status, it depends from the stability and the nature of your protein (e.g it contains structural disulfide bondes) because the precence of SDS in the loading and running buffer in SDS-page in principle make this kind of gel denaturing.
Practically some proteins, that thanks to the presence of S-S bonds or other factors are very stable are able to mantain their native conformation also when loaded in SDS-page w/o addiction of reducing agent in the laemil buffer and avoiding the sample boiling step. Protein with stable disulfide are able to resist also to the boiling step in absence of reducing agent, but this result cannot be generalized.
For example in the attached slide you can see the SDS-page of 3 proteins. One of this, that contain a structural disulfide bonds, show difference in RUN w and w/O reducing agents while for the other 2 no main differences where obseved suggesting that there are not present structural S-S bonds and that probably boiling 5' at 95°C whas able to completelly denature the protein also in absence of reducing agent.
Some times for some proteins, in presence and absence of recuging agent you can observe also the opposite behaviours that means that protein with DTT run at lower molecular weight respect protein w/o DTT. In this case you probably are in presence of intermolecular S-S bonds which lead to the formation of protein aggregates.
SDS is an amphipathic surfactant. It denatures your proteins by binding to the protein chain with its hydrocarbon tail, exposing normally buried regions .
I would just like to clarify what I think you are asking:
SDS is an negatively charged detergent used in SDS PAGE, usually combined with heat to denature proteins and give them the same charge to mass ratio so proteins separate on the basis of size. If you do not heat your sample but still have SDS present (as I think you are suggesting), depending on the protein, your samples could run differently because the whole protein may not be adequately denatured just by the presence of SDS alone i.e. it will have more secondary structure and therefore be slightly more compact, and will potentially run faster. SDS will break ionic interactions so may give you information about the nature of the protein-protein interactions.
Other ways of separating proteins are under denaturing (SDS/Heat) and reducing conditions. The presence of the reducing agent breaks any disulphide bonds and unfolds the protein even further breaking secondary structures formed by these bonds. Proteins treated in this way will tend to run differently to non reduced samples and is a technique used to determine if higher order structures are linked by S-S bonds.
Running gels without heat/SDS is called native-PAGE in which proteins retain their overall charge, shape (and size) and therefore separate close to their "native structure" : monomer, dimer, trimer etc