Axial Chirality in Carbodiimides
Yes, carbodiimides exhibit axial chirality similar to allenes. While there are significant differences in the allene - carbodiimide systems (see below) there are also some similarities. For example, just like in amines and imines the barrier to racemization in generally carbodiimidesis low, typically in the 6-10 kcal/m range. 

Anet did a study on 1,3-diisopropylcarbodimide ("Energy barrier of racemization in diisopropylcarbodiimide", Frank A. L. Anet, J. C. Jochims, C. H. Bradley
J. Am. Chem. Soc., 1970, 92 (8), pp 2557–2558, DOI: 10.1021/ja00711a064).



In the chiral compound each isopropyl group would contain two diastereotopic methyl groups split by the isopropyl proton so we would expect to see 2 doublets. On the other hand, if racemization is rapid all methyl groups become equivalent and we would then expect to see only one doublet. Experimentally it turns out that around -150 C the doublet broadens into 2 doublets giving rise to a barrier height for racemization around 6.7 kcal/m.

Important Notes:

1) Just like with amines and imines, placing very electronegative groups on the carbodiimide nitrogens (fluorine, for example) raises the barrier to racemization.  Incorporating the carbodiimide structure into a small ring (see here for an example) can also raise the barrier to enatiomerization.  Both techniques have been used to permit isolation of chiral carbodiimides.

2) Carbodiimides are not linear, but rather the $\ce{N-C-N}$ angle is around 170°.  This is attributed to the terminal nitrogens being more electronegative than the central carbon and consequently affecting the hybridization of the central carbon (e.g. changing it from $\ce{sp^1}$ to $\ce{sp^{x}}$ where 1            &lt; x &lt; 2)                 

3) Racemization in carbodiimides can be achieved by two different mechanisms


by traditional nitrogen inversion involving a planar nitrogen atom, and/or
by traditional cis-trans isomerization about the $\ce{C=N}$ double bond


both mechanisms are thought to play a role in carbodiimide racemization.