The answer, I believe, is contained in the verse itself. The verse does not refer to Moses, but to God. There may be a little confusion in that some manuscripts read "Christ", some read "God", and still others read "the Lord". The majority of manuscripts have "the Lord".
The event behind those who tempted and were destroyed by serpents is described in Numbers 21:5ff:
And the people spoke against God and against Moses, saying, Why is this? Hast thou brought us ought of Egypt to slay us in the wilderness? for there is not bread nor water; and our soul loathes this light bread. And the Lord sent among the people deadly serpents, and they bit the people, and much people of the children of Israel died. And the people came to Moses and said, We have sinned, for we have spoken against the Lord, and against thee: pray therefore to the Lord, and let him take away the serpent from us.
The word translated by the NIV as "tempt" is πειράζω (peirazō) which really means to "test" (the RSV reads We must not put the Lord to the test). The NIV itself translates πειράζω as "test" in other places:
The Pharisees came and began to question Jesus. To test him, they asked him for a sign from heaven (Mark 8:11)
Jesus answered, “It is said: ‘Do not put the Lord your God to the test.’” (Luke 4:12)
The "testing" (or "tempting") referred to is failure to trust in God in adversity, and instead murmuring against Him.
