[a synthesis of Evangelos, Michael and my comments / posts]
You are right if you have a simple, two-time period diff-in-diff and you believe parallel trends is true OR if you have staggered/heterogenous timing with comparator group only being the never treated.
But, if we sometimes compare the treated with different groups of pre-treated (as in staggered/heterogenous timing with comparator group including the pre-treated), then we could get a negative coefficient with anticipation effects.
As a motivating example, say we're evaluating effects of cash transfers, and we only have two time periods. Some (very poor individuals) anticipate they will receive a cash transfer, and therefore spend before receiving the cash transfer to help improve their livelihoods. Others do not anticipate and spend to a lesser degree after receiving the cash transfer. Then, since treatment timing is homogeneous (rich and poor treated at the same time), the diff in diff is:
[Average of treated post] - [Average of treated pre] - ([Average of control post] - [Average of control pre])
The treated comprise both rich and poor. Anticipatory behavior generally means that [Average of treated pre] is larger.
However, you'd get a negative effect if somehow there was heterogeneity in treatment timing. Suppose instead that the rich were treated first, while the poor were untreated. The poor anticipated treatement, and increased spending while untreated. Then the comparison between treated and pre-treated will result in the pre-treated having increased spending a lot (due to anticipation) and a potentially negative estimate. This gets resolved if you restrict the comparator group to the never-treated.
I'd still be wary, however. It is possible that anticipatory behavior means that [Average of treated pre] is smaller (take moral hazard with health insurance, for example).
Moreover, treatment effects can incorporate (1) a change in levels and (2) a change in trends. If treatment (and anticipation of treatment), causes a change in trends, then parallel trends will fail, causing problems. In general, pre-trends due to anticipatory behavior can't be distinguished from pre-trends due to any other reason, which would imply a failure of parallel trends.
Also, nice review article of the literature arxiv.org/abs/2201.01194.