Which random effects to include in this GLMM? In my study growth of plants was measured in different years on different plots (all plants were measured in all years).

The question I'd like to answer with my model is: Which factors influence growth and how? 
I do not want to make predictions on growth, I am simply interested in the effects of the predictors. I am also not interested in overall fixed effects of year or plot themselves.
My predictors can be grouped in three categories:


*

*Predictors which characterise the sampled plants (these are
continuous or categorical): A1, A2, A3 (4 levels)

*Predictors which characterise the plot (these are continuous): B1, B2

*Predictors which characterise the conditions in the years of sampling (these are continuous): C1, C2


I am so far using a GLMM with the different predictors as fixed effects and plant ID as a random effect. 
growth ~ A1 + A2 + A3 + B1 + B2 + C1 + C2 + (1|Plant ID)


However, I am wondering if I have to include year of sampling and plot as additional random effects? This would especially change the standard errors of predictor A3 a lot. The categorical predictor A3 is a bit special as it has four levels which change between years but within each year each plant across all plots has the same level.
growth ~ A1 + A2 + A3 + B1 + B2 + C1 + C2 + (1|Plant ID) + (1|Year)


.. which is the same as:
growth ~ A1 + A2 + A3 + B1 + B2 + C1 + C2 + (1|Plot:Plant ID) + (1|Year)


.. but different from:
growth ~ A1 + A2 + A3 + B1 + B2 + C1 + C2 + (1|Plant ID) + (1|Year) + (1|Plot)


Do you have any advice for me on how to correctly specify the random effects?
Why does adding Plot and Year as random effects increase the standard errors of the estimates of (some) fixed effects?
 A: From the picture and descriptions you give, I think Plant is nested within Plot and both of them are crossed with Year of sampling.
If this is the case, then you could account for the dependencies in your data with a model like
growth ~ A1 + A2 + A3 + B1 + B2 + C1 + C2 + (1 | Plot/Plant) + (1 | Year)

Whether you should include Plot and Year of sampling as random grouping factors depends on how much variance there is between the different levels within that grouping factors. Personally, I would start with the maximal random effects structure justified by the design and reduce the model if it turns out to be overparameterized (see e.g. Bates et al., 2015 or this answer).
(1 | Plot:Plant) + (1 | Year) is the same as (1 | Plant) + (1 | Year) because Plant is nested explicitly within Plot, i.e., the levels of Plant are coded uniquely across the levels of Plot. 

Why does adding Plot and Year as random effects increase the standard
  errors of the estimates of (some) fixed effects?

This is because all your fixed predictors are between-unit-predictors (meaning they vary between certain levels of a random grouping factor, not within). Thus, intuitively speaking, if the data points of different plants, plots or years vary a lot (and independent of the factor levels or range of the fixed effects), the effects of the predictors are less reliable - and this is reflected by the increased standard errors.
