Why does lithium forms no ethynide on reacting with ethyne?
Sodium alkynide was formed by reacting sodium and acetylene but in case of lithium alkynide, it can be formed by reacting acetylene with organolithium reagents like BuLi, Lithium amide, LiHMDS. From wikipedia:


  Lithium amide, LiHMDS, or organolithium reagents, such
  as butyllithium, are frequently used form lithium acetylides:
  
  $\ce{H-C#C-H + BuLi -&gt;[THF][-78℃] Li -#-H + BuH}$


The above process is well described in this paper which @Tyberius linked in the comment section.