By RAW, it isn't possible to walk through a Flaming Sphere, as the spell creates a semi-solid spongy flaming object. However, normal clothes would burn just fine.. if it weren't for a few.. issues.
By RAW, energy damage such as fire, lightning, or cold deals half damage to objects, so hardness is calculated after the damage is divided by two.
Energy attacks deal half damage to most objects. Divide the damage by 2 before applying the object's hardness. Some energy types might be particularly effective against certain objects, subject to GM discretion. For example, fire might do full damage against parchment, cloth, and other objects that burn easily. Sonic might do full damage against glass and crystal objects.
This falls under the "objects that might burn easily" so we'll move onto the next step.
Attended (Held/Wielded etc.) Items: Unless the descriptive text for a spell (or attack) specifies otherwise, all items carried or worn by a creature are assumed to survive a magical attack. If a creature rolls a natural 1 on its saving throw against the effect, however, an exposed item is harmed (if the attack can harm objects). Refer to Table: Items Affected by Magical Attacks to determine order in which items are affected. Determine which four objects carried or worn by the creature are most likely to be affected and roll randomly among them. The randomly determined item must make a saving throw against the attack form and take whatever damage the attack dealt. If the selected item is not carried or worn and is not magical, it does not get a saving throw. It simply is dealt the appropriate damage.
If you are the target of a spell or a spell-like ability and you fail your save with a
1, your items become subject to magical damage. Those items are chosen from the list in the order below:
Table: Items Affected by Magical Attacks
3rd Magic helmet, hat, or headband
4th Item in hand (including weapon, wand, or the like)
5th Magic cloak
6th Stowed or sheathed weapon
7th Magic bracers
8th Magic clothing
9th Magic jewelry (including rings)
10th Anything else
Your vestments aren't magical, so they would fall under the category of "Anything else". If you weren't wearing any items in your other slots, they would be affected long before your clothes would be burned away. However, if you didn't have anything in those other slots.. Your clothes would likely be burnt to a cinder.. on one side anyway, provided all of the prior things were to have been met.
If that was the case, damage would be calculated normally, minus your fire resistance, and any leftover damage would be applied to you, and to your clothing...
And since the damage rolled was lower than your fire resistance, you would take no damage, and items you wear would take no damage from the Flaming sphere.