Sorry for the belated answer - but yes, it is inevitable. Black body radiation means everything in the universe glows.
This whole thing is just a problem with an universe both infinite and old, though. In a young universe, the dust might still be cold enough (yet still hotter than it is in reality), and in a finite universe, there could be a steady state as you described.
Better still, in a young expanding universe (that still may yet some day contract) the visible horizon includes less than it once did as the mean distance between matter has increased.
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Sorry for the belated answer - but yes, it is inevitable. Black body radiation means everything in the universe glows.
This whole thing is just a problem with an universe both infinite and old, though. In a young universe, the dust might still be cold enough (yet still hotter than it is in reality), and in a finite universe, there could be a steady state as you described.
Better still, in a young expanding universe (that still may yet some day contract) the visible horizon includes less than it once did as the mean distance between matter has increased.