The origin of interstellar dust in galaxies is poorly understood, particularly the relative contributions from supernovae and the cool stellar winds of low-intermediate-mass stars. Recently, large masses of newly formed dust have been discovered in the ejecta of core-collapse supernovae. Here, we present Herschel Photodetector Array Camera and Spectrometer (PACS) and Spectral and Photometric Imaging Receiver (SPIRE) photometry at 70–500 m of the historical, young supernova remnants: Kepler and Tycho, both thought to be the remnants of Type Ia explosion events. We detect a warm dust component in Kepler’s remnant with and mass ; this is spatially coincident with thermal X-ray emission and optical knots and filaments, consistent with the warm dust originating in the circumstellar material swept up by the primary blast wave of the remnant. Similarly for Tycho’s remnant, we detect warm dust at with mass . Comparing the spatial distribution of the warm dust with X-rays from the ejecta and swept-up medium, and Hα emission arising from the post-shock edge, we show that the warm dust is swept up interstellar material. We find no evidence of a cool (25–50 K) component of dust with mass ≥0.07 M⊙ as observed in core-collapse remnants of massive stars. Neither the warm or cold dust components detected here are spatially coincident with supernova ejecta material. We compare the lack of observed supernova dust with a theoretical model of dust formation in Type Ia remnants which predicts dust masses of 88(17) × 10−3 M⊙ for ejecta expanding into ambient surrounding densities of 1(5) cm−3. The model predicts that silicon- and carbon-rich dust grains will encounter, at most, the interior edge of the observed dust emission at ∼400 years, confirming that the majority of the warm dust originates from swept-up circumstellar or interstellar grains (for Kepler and Tycho, respectively). The lack of cold dust grains in the ejecta suggests that Type Ia remnants do not produce substantial quantities of iron-rich dust grains and has important consequences for the ‘missing’ iron mass observed in ejecta. Finally, although, we cannot completely rule out a small mass of freshly formed supernova dust, the Herschel observations confirm that significantly less dust forms in the ejecta of Type Ia supernovae than in the remnants of core-collapse explosions.