The effects of oral hyaluronic acid on intraocular pressure are highly individual, according to anecdotal reports from FitEyes members who monitor their IOP and have tracked their use of hyaluronic acid or glucosamine or chondroitin sulfate.
However, in general, the following publication indicates that steroid-free hyaluronic acid injections in knee are expected to be free of IOP side effects.
CORR Insights®: Intraocular Pressure Increases After Intraarticular Knee Injection With Triamcinolone but Not Hyaluronic Acid
In the current study, Taliaferro and colleagues  show that all interventions, even seemingly low-risk corticosteroid injections for knee arthritis, can have unanticipated side effects. The well-known risks of intraarticular corticosteroid injection include infection, bleeding, steroid flare reaction, glycemic lability in patients with diabetes, skin discoloration, and skin/subcutaneous fat atrophy . Although intraocular pressure elevation is not a well-described side effect of intraarticular corticosteroid injection, Taliaferro and colleagues found that nearly 30% of patients undergoing intraarticular knee corticosteroid injections demonstrate increased intraocular pressure (> 7 mm Hg) at 1 week after the injection, which remains elevated in about half of those patients at 1 month .
Prior studies have shown that steroid-induced intraocular pressure elevations may take 10 days to 14 days to manifest, and so by measuring intraocular pressure at 7 days, it is possible that the current study design missed additional steroid responders.
Below is the result that is relevant to this question:
Results: The mean intraocular pressure increased by 2.79 mm Hg 1 week after treatment with triamcinolone, but it did not change among those patients treated with hyaluronic acid (2.79 ± 9.9 mm Hg versus -0.14 ± 2.96 mm Hg; mean difference 2.93 mm Hg; 95% confidence interval, -0.71 to 6.57 mm Hg; p = 0.12).
More patients who received triamcinolone injections developed an increase in intraocular pressure > 7 mm than did those who received hyaluronic acid (29% [nine of 29] versus 0% [zero of 31]; p = 0.002).
The paragraph above points out that 0 of the patients in this study showed IOP increase after steroid-free hyaluronic acid injections in knee.
The limitations are that it is a small study and they may have missed some responders due to checking at 7 days but not 10 to 14 days after injection. However, as the article points out, with no reports of problems in the medical literature, "one can assume the risk is low."