We don't know, but some believed that Saruman did discover his remains.
But King Elessar, when he was crowned in Gondor, began the re-ordering of his realm, and one of his first tasks was the restoration of Orthanc, where he proposed to set up again the palantir recovered from Saruman. Then all the secrets of the tower were searched. Many things of worth were found, jewels and heirlooms of Eorl, filched from Edoras by the agency of Wormtongue during King Thĕoden's decline, and other such things, more ancient and beautiful, from mounds and tombs far and wide. Saruman in his degradation had become not a dragon but a jackdaw. At last behind a hidden door that they could not have found or opened had not Elessar had the aid of Gimli the Dwarf a steel closet was revealed. Maybe it had been intended to receive the Ring; but it was almost bare. In a casket on a high shelf two things were laid. One was a small case of gold, attached to a fine chain; it was empty, and bore no letter or token, but beyond all doubt it had once borne the Ring about Isildur's neck. Next to it lay a treasure without price, long mourned as lost for ever: the Elendilmir itself, the white star of Elvish crystal upon a fillet of mithril that had descended from Silmarien to Elendil, and had been taken by him as the token of royalty in the North Kingdom. Every king and the chieftains that followed them in Arnor had borne the Elendilmir down even to Elessar himself; but though it was a jewel of great beauty, made by Elven-smiths in Imladris for Valandil Isildur's son, it had not the ancientry nor potency of the one that had been lost when Isildur fled into the dark and came back no more. (Unfinished Tales)
Since these were among Isildur's personal effects, it was suspected that Saruman had found his remains, taken the treasures, and dishonored his bones.
When men considered this secret hoard more closely, they were dismayed. For it seemed to them that these things, and certainly the Elendilmir, could not have been found, unless they had been upon Isildur's body when he sank; but if that had been in deep water of strong flow they would in time have been swept far away. Therefore Isildur must have fallen not into the deep stream but into shallow water, no more than shoulder-high. Why then, though an Age had passed, were there no traces of his bones? Had Saruman found them, and scorned them - burned them with dishonour in one of his furnaces? If that were so, it was a shameful deed; but not his worst.
As far as I know, these are the only reference to his remains.
Personally I find it very doubtful human remains would have lasted that long sitting at the bottom of a swampy river, but I suppose I must defer to the judgement of the Wise who would not have been afraid that Saruman found the remains if that was a possibility.