Savings would be means-tested for Rent Supplement. Here is how it's assessed (taken from
www.welfare.ie)
Savings Means assessed (weekly)
First €5,000 Nil
Next €10,000 €1 per €1,000
Next €25,000 €2 per €1,000
Over €40,000 €4 per €1,000
So, for example, if you had €15000 saved, the weekly means would be €10 per week.
If you are on a means-tested Social Welfare Payment (e.g. Jobseeker's Allowance, Disability Allowance, One-parent Family Payment) that payment could be affected by savings as well (although the means assessed may be different to Rent Supplement).