You have to use strftime function for your date series of dataframe to plot xtick labels correctly:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from datetime import datetime, timedelta
import random
dates = [datetime.today() - timedelta(days=x * random.getrandbits(1)) for x in xrange(25)]
df = pd.DataFrame({'depth': [0.1,0.05, 0.01, 0.005, 0.001, 0.1, 0.05, 0.01, 0.005, 0.001, 0.1, 0.05, 0.01, 0.005, 0.001, 0.1, 0.05, 0.01, 0.005, 0.001, 0.1, 0.05, 0.01, 0.005, 0.001],
'date': dates,
'value': [-4.1808639999999997, -9.1753490000000006, -11.408113999999999, -10.50245, -8.0274750000000008, -0.72260200000000008, -6.9963940000000004, -10.536339999999999, -9.5440649999999998, -7.1964070000000007, -0.39225599999999999, -6.6216390000000001, -9.5518009999999993, -9.2924690000000005, -6.7605589999999998, -0.65214700000000003, -6.8852289999999989, -9.4557760000000002, -8.9364629999999998, -6.4736289999999999, -0.96481800000000006, -6.051482, -9.7846860000000007, -8.5710630000000005, -6.1461209999999999]})
pivot = df.pivot(index='depth', columns='date', values='value')
sns.set()
ax = sns.heatmap(pivot)
ax.set_xticklabels(df['date'].dt.strftime('%d-%m-%Y'))
plt.xticks(rotation=-90)
plt.show()