A Status Orange rain warning has been issued for Wicklow and Waterford, and a Yellow alert for nine other countries, with more flooding likely.
The Yellow warning is for Carlow, Dublin, Kildare, Kilkenny, Laois, Louth, Wexford, Monaghan and Tipperary.
Both will come into effect at 3am tomorrow and last for 24 hours.
Met Éireann said that heavy rain falling on saturated ground, combined with high river levels and high tides, will lead to localised and river flooding, as well as difficult travel conditions.
Motorists in the southeast are being warned not to drive into flood water.
Wexford County Council said that it is "dangerous and unpredictable, and even shallow water can hide strong currents, debris and open drains".
It said that roads are flooded in places where motorists might not ordinarily expect flooding.
The council is installing aqua dams along exposed parts of the quay in Wexford town ahead of high tide tomorrow morning.
The council said that much of the spot flooding and pooling of surface water is caused by run-off from land, which combined with full drainage systems, will take time to recover.
High coastal levels will reduce the capacity for river discharge and increase flood risk in low-lying areas, especially at high tide, it said.
The council added that high spring tides for the remainder of the week, combined with strong winds and surge, may cause wave overtopping and coastal flooding in places.
Watch: Drone footage shows scale of Enniscorthy flooding
Wicklow County Council said its teams are monitoring the situation and identifying potential impacts in the county.
It said that requests for sandbags are subject to an assessment of need the availability of stock.
The council urged people to stay away from coastal edges, noting that road closures are in place in certain areas affected by previous flooding.
Dublin, southeast at high risk for flooding
Parts of Dublin and the southeast of the country are at high risk of flooding, according to the National Director for Fire and Emergency Management.
Speaking on RTÉ's Morning Ireland, Keith Leonard said the key time will be tomorrow night.
"The Nore, the Barrow, the Slaney and the Liffey [river] catchment are going to see very high levels right across this evening and into tomorrow," he said.
Mr Leonard added that local authorities are sharing resources and equipment is being moved from less affected areas to key pinch points, with interim solutions, such as an aqua dam in Enniscorthy, being put in place.
The town was one of a number of areas to be hit by heavy flooding last week after Storm Chandra hit.
Parts of south Dublin, including Rathfarnham, were also badly affected after the River Dodder burst its banks.
Mr Leonard said that "absolutely every engineering solution and every kind of interim measure that can be taken is being taken to try and deal with these almost record-breaking levels of water right across those catchments".
In relation to farming, he said: "This is about as bad a weather as you could have for that type of situation.
"Hopefully there'll be some respite shortly, but no doubt, extremely difficult conditions across the agriculture sector."
'Risk of flooding doesn't stop because rain has stopped'
Met Éireann meteorologist Andrew Doran Sherlock warned that even when it is not raining, there can be delayed responses in some areas.
"The risk of flooding doesn’t necessarily stop because the rain has stopped", he said, adding that further precipitation could lead to larger impacts.
With strengthening easterly winds tomorrow, there is a risk of coastal flooding that could lead to further impacts, Mr Doran Sherlock said.
"People really need to pay attention to what the weather conditions are, and what river gauge data is, and you can check that on waterlevel.ie."
Local authorities will have the best guidance as they will have a good idea of response times in the areas, he added.
Mr Doran Sherlock said the rain may ease on Saturday, and Sunday is looking likely to be the driest day of the week, but next week, the weather will still be dominated by low pressure so there "isn't really a let up".
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Tánaiste and Minister for Finance Simon Harris said there will be further engagement on the flood warning system and "what level of information have been put out into the public domain to help communities".
"We will obviously increase the level of financial assistance available to businesses.
"We also had a very important conversation about what interim works can take place.
"I think we can very clearly see there's significant difference between the Yellow weather warning for rain when the rivers are low versus the Yellow weather warning when the rivers are high," Mr Harris said.
Additional reporting Petula Martyn
