The accepted answer is very good. I thought it might be helpful to supplement that answer with a comment about the concentration of sodium in an oral rinse, along with an actual BP (blood pressure) measurement comparison.
Recommendations for saline mouthwashes include 0.9% sodium chloride followed by pure water rinse, a home recipe for this solution being 1 teaspoon of salt added to a pint (2 cups) of water Mouth care guidelines.
I typically use a far higher concentration apparently, about a teaspoon of salt in 1/3 cup. This almost certainly results in too much sodium absorption through the oral mucosa. Earlier today my BP (brachial cuff Omron BP710 automatic device) was 113/75 (systolic/diastolic mm Hg) at 78 BPM (heart rate beats/min) prior to oral saline rinse and 127/81 at 80 BPM following the saline rinse.
To make comparisons of the measurements easier, I calculate PP (pulse pressure), which is systolic pressure minus diastolic. Pre-saline rinse PP then 38 vs post-saline PP 46.
Also useful is calculating MAP (mean arterial pressure estimated) Mean arterial pressure My usual method of calculating MAP using brachial cuff pressure reading is take a third of the PP (pulse pressure/3), add that to the DP (diastolic pressure) reading, and add a correction term of 5 mm Hg to that result. My MAP pre-saline is then 93 and post-saline 101.
On the basis of my "observational study" I conclude that there is definitely the possibility of increasing systemic blood pressure by oral rinsing with saline solution. However, using more dilute saline would of course reduce the observed BP elevation.